Doubleintegral calculator

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doubleintegral calculator

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What is a double integral calculator and how to use it?

Answer:

A double integral calculator is an online or software tool designed to evaluate double integrals, which are used to integrate functions of two variables over a two-dimensional region. Double integrals are extensively used in mathematics, physics, and engineering to find areas, volumes, mass, probabilities, and more when dealing with functions defined in two dimensions.

Table of Contents

  1. What is a Double Integral?
  2. Purpose of Double Integral Calculations
  3. How the Double Integral Calculator Works
  4. Input Requirements for the Calculator
  5. Step-by-Step Example of Double Integral Calculation
  6. Summary Table: Key Concepts

1. What is a Double Integral?

A double integral is an extension of a single integral to two variables. If you have a function f(x, y) defined over a region D in the xy-plane, the double integral of f over D is generally written as:

\iint_D f(x,y) \, dA

where dA represents an infinitesimal area element on the plane (usually dx dy or dy dx). It sums values of the function over the region, calculating for example:

  • The volume under the surface (z = f(x,y)) above the region (D).
  • The mass if the function represents density.
  • The area of a region when (f(x,y) = 1).

2. Purpose of Double Integral Calculations

Double integrals have several practical uses:

  • Area of a planar region when the function equals 1.
  • Volume under a surface.
  • Center of mass and moments of inertia by integrating weighted density functions.
  • Probability in joint distributions.
  • Flux calculations in vector calculus.

3. How the Double Integral Calculator Works

A double integral calculator typically allows users to:

  • Input the function (f(x,y)) to be integrated.

  • Specify the region limits for (x) and (y). This can be rectangular (simple limits) or more complex (like (y) limits depending on (x)).

  • The calculator then performs the integration, often by:

    • Evaluating the inner integral with respect to one variable.
    • Integrating the resulting expression with respect to the other variable.

Mathematically, a double integral over region (D) can be written as an iterated integral:

\int_{a}^{b} \int_{c(x)}^{d(x)} f(x,y) \, dy \, dx

where the boundary functions (c(x)) and (d(x)) define the (y)-limits for each fixed (x).

The calculator automates this computation using symbolic or numeric integration methods.

4. Input Requirements for the Calculator

When using a double integral calculator (such as the one from the provided forum), you typically need to:

  • Enter the function (f(x,y)) in proper syntax (e.g., x*y + y^2 or sin(x)*cos(y)).
  • Define the integration limits:
    • For rectangular regions: enter numeric constants for (x) and (y) limits, such as (x:0 \to 2), (y:1 \to 3).
    • For more complex regions: input the limits as functions, e.g., (y: 0 \to x^2).
  • Optionally, select the order of integration — integrate first with respect to (x) or (y).
  • Click Calculate to obtain the value of the integral with step-by-step details, if supported.

5. Step-by-Step Example of Double Integral Calculation

Let’s consider the example:

Calculate the double integral of (f(x,y) = x + y) over the rectangular region (0 \leq x \leq 1), (0 \leq y \leq 2).

Step 1: Write the integral

\int_0^1 \int_0^2 (x + y) \, dy \, dx

Step 2: Perform inner integral over (y)

For fixed (x),

\int_0^2 (x + y) \, dy = \left[ xy + \frac{y^2}{2} \right]_0^2 = x(2) + \frac{2^2}{2} - 0 = 2x + 2

Step 3: Perform outer integral over (x)

\int_0^1 \left(2x + 2\right) dx = \left[ x^2 + 2x \right]_0^1 = 1 + 2 - 0 = 3

Result: The value of the double integral is 3.

A double integral calculator automates this process and returns the result, often showing intermediate steps like above for better understanding.

6. Summary Table: Key Concepts

Concept Definition Example
Double Integral Integral of a 2-variable function over a 2D region (\iint_D f(x,y) dA)
Iterated Integral Expressing double integral as nested single integrals (\int_a^b \int_{c(x)}^{d(x)} f(x,y) dy dx )
Integration Limits Boundaries for the variables in the double integral (x: 0 \to 1, y: 0 \to 2)
Purpose Finding volume, area, mass, probability, etc. Calculating mass from a density function
Calculator Input Function (f(x,y)), integration limits, order Function: (x + y), Limits: (0 \to 1), (0 \to 2)

Summary

A double integral calculator efficiently evaluates integrals involving two variables over a specific region, enabling fast and accurate solution of problems in calculus and applied sciences. You simply enter the function and limits, and it computes the double integral, often with step-by-step details for your learning. This is especially useful for complex regions or functions where manual integration is laborious.

If you want, I can help you with a specific double integral example or guide you on how to use such a calculator on any platform.

@Dersnotu

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Doubleintegral calculator

Answer:

A double integral calculator is a tool used to compute double integrals, which are mathematical operations for finding areas, volumes, and other quantities in two-dimensional spaces. Double integrals are a key concept in calculus, particularly in multivariable calculus, and are often used to solve problems involving surfaces and regions in the plane. In this response, I’ll explain what double integrals are, how to compute them step by step, and how you can use online calculators to simplify the process. I’ll also provide examples, applications, and a summary to help you understand this topic thoroughly.

Since your query is in English, I’ll respond in English, keeping the explanation clear, accurate, and tailored for students or anyone learning calculus. Let’s break this down step by step, using simple language and including visual aids where possible.

Table of Contents

  1. What is a Double Integral?
  2. Key Terminology
  3. Step-by-Step Computation of Double Integrals
  4. Online Double Integral Calculators
  5. Examples and Applications
  6. Summary Table
  7. Final Summary

1. What is a Double Integral?

A double integral is an extension of the single integral you might know from basic calculus. While a single integral calculates areas under curves in one dimension, a double integral computes quantities like volumes under surfaces or areas of regions in two dimensions. It’s represented mathematically as:

\iint_R f(x,y) \, dA

where:

  • f(x,y) is the function being integrated (e.g., a surface height).
  • R is the region in the xy-plane over which the integration occurs.
  • dA is the area element, often written as dx dy or dy dx, depending on the order of integration.

Double integrals are used in various fields, such as physics for calculating mass or center of mass, engineering for stress analysis, and economics for optimizing regions. They can be iterated integrals, where you integrate with respect to one variable first and then the other, or they can use more advanced techniques like changing to polar coordinates for circular regions.

For students, think of a double integral as a way to “sum up” values over a 2D area, similar to how you add up pixels in an image to find total brightness.

2. Key Terminology

To make this topic easier to grasp, here are some important terms defined simply:

  • Function (f(x,y)): A mathematical expression that depends on two variables, x and y. For example, f(x,y) = x^2 + y^2 describes a paraboloid surface.
  • Region (R): The area in the xy-plane where the integration happens. It could be a rectangle, circle, or irregular shape, defined by boundaries like 0 ≤ x ≤ 2 and 0 ≤ y ≤ 3.
  • Iterated Integral: A double integral split into two single integrals, e.g., $$ \int_a^b \int_c^d f(x,y) , dy , dx $$. The order (dx dy or dy dx) matters and can affect ease of computation.
  • Limits of Integration: The bounds for x and y that define the region R. For instance, in a rectangular region, you might have lower and upper limits for each variable.
  • Area Element (dA): Represents an infinitesimal piece of area in the plane, analogous to dx in single integrals.
  • Double Integral Calculator: A digital tool (often online) that automates the computation of double integrals by inputting the function and region boundaries.

Understanding these terms helps in visualizing and solving double integral problems without getting overwhelmed.

3. Step-by-Step Computation of Double Integrals

Computing a double integral manually involves breaking it down into smaller steps. I’ll use an example to demonstrate, as this is a numerical question. Let’s solve the double integral $$ \iint_R (x + y) , dA $$ over the rectangular region R where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. This is a simple case, but the process scales to more complex problems.

Step 1: Set Up the Integral

  • Identify the region R and the function. Here, R is a square from (0,0) to (1,1), and f(x,y) = x + y.
  • Write the double integral as an iterated integral. Since the region is rectangular, we can use:
\int_0^1 \int_0^1 (x + y) \, dy \, dx

(We’re integrating with respect to y first, then x.)

Step 2: Compute the Inner Integral

  • Focus on the inner integral with respect to y, treating x as a constant:
\int_0^1 (x + y) \, dy
  • Integrate term by term:
  • The integral of x (constant with respect to y) is x*y.
  • The integral of y is (1/2)y^2.
  • Evaluate from y = 0 to y = 1:
\left[ x \cdot y + \frac{1}{2} y^2 \right]_0^1 = \left( x \cdot 1 + \frac{1}{2} \cdot 1^2 \right) - \left( x \cdot 0 + \frac{1}{2} \cdot 0^2 \right) = x + \frac{1}{2}

Step 3: Compute the Outer Integral

  • Now integrate the result with respect to x:
\int_0^1 \left( x + \frac{1}{2} \right) \, dx
  • Integrate term by term:
  • The integral of x is (1/2)x^2.
  • The integral of 1/2 is (1/2)x.
  • Evaluate from x = 0 to x = 1:
\left[ \frac{1}{2} x^2 + \frac{1}{2} x \right]_0^1 = \left( \frac{1}{2} \cdot 1^2 + \frac{1}{2} \cdot 1 \right) - \left( \frac{1}{2} \cdot 0^2 + \frac{1}{2} \cdot 0 \right) = \frac{1}{2} + \frac{1}{2} = 1

Step 4: Interpret the Result

  • The value of the double integral is 1. This represents the volume under the surface z = x + y over the unit square region.

For non-rectangular regions or more complex functions, you might need to change the order of integration, use substitution, or switch to polar coordinates. For example, if R is a circle, polar coordinates can simplify the limits.

If you’re dealing with a specific double integral, you can input it into a calculator for quick results, which I’ll cover next.

4. Online Double Integral Calculators

Manually computing double integrals can be tedious, especially for irregular regions or complex functions. Fortunately, several online tools and software can handle this for you. These calculators use numerical methods (like Simpson’s rule or Monte Carlo simulations) or symbolic computation to provide exact or approximate results.

Popular Online Tools

  • Wolfram Alpha: A powerful computational engine that can evaluate double integrals symbolically or numerically. For example, input “double integral of (x+y) dx dy from x=0 to 1, y=0 to 1” and it will return the result instantly. It’s free for basic use and cites sources from mathematical databases.
  • Symbolab: User-friendly with step-by-step solutions. Enter the function and limits, and it shows the iterated integral process, similar to what I did above. It’s great for learning as it explains each step.
  • Integral Calculator ( ): Specifically designed for integrals, it supports double and triple integrals. You can input the function and region, and it provides both the answer and a graph.
  • Desmos: More of a graphing calculator, but it can visualize double integrals by plotting the surface and region. It’s interactive and free, helping you see the geometry before computing.
  • MATLAB or Python (via libraries): For advanced users, tools like MATLAB’s integral2 function or Python’s SciPy library can compute double integrals programmatically. This is useful for custom regions or large datasets.

How to Use a Double Integral Calculator

  1. Input the Function: Enter f(x,y), e.g., “x+y”.
  2. Define the Region: Specify the limits, like “x from 0 to 1, y from 0 to 1”.
  3. Choose Integration Type: Some calculators allow you to select rectangular, polar, or other coordinate systems.
  4. Run the Calculation: The tool computes the result and may provide a graph or steps.
  5. Verify the Output: Always check the result by understanding the context, as numerical methods can have small errors.

These tools are based on up-to-date algorithms and are reliable for educational and professional use. For instance, Wolfram Alpha uses data from the Wolfram Language, which is continually updated with the latest mathematical research.

To enhance your understanding, I’ll now call a function to generate an image illustrating a double integral example. This will provide a visual representation of the region and surface.

5. Examples and Applications

Double integrals have wide-ranging applications. Here are a few examples to make the concept more relatable:

Example 1: Finding Volume

Suppose you want to find the volume under the surface z = x^2 + y^2 over the region R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.

  • Set up the integral: $$ \iint_R (x^2 + y^2) , dA = \int_0^1 \int_0^1 (x^2 + y^2) , dy , dx $$
  • Compute step by step (similar to earlier):
  • Inner integral: $$ \int_0^1 (x^2 + y^2) , dy = \left[ x^2 y + \frac{1}{3} y^3 \right]_0^1 = x^2 + \frac{1}{3} $$
  • Outer integral: $$ \int_0^1 \left( x^2 + \frac{1}{3} \right) , dx = \left[ \frac{1}{3} x^3 + \frac{1}{3} x \right]_0^1 = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} $$
  • Result: The volume is 2/3. Using a calculator like Wolfram Alpha would give the same answer quickly.

Example 2: Area of a Region

If f(x,y) = 1, the double integral over R gives the area of R. For a circle of radius 1, use polar coordinates:

  • $$ \iint_R 1 , r , dr , d\theta $$ from r=0 to 1, θ=0 to 2π.
  • Result: Area = π, which matches the formula for a circle.

Applications

  • Physics: Calculate the center of mass of a lamina (thin plate) with varying density.
  • Economics: Find total cost or profit over a region, e.g., integrating demand functions over price and quantity.
  • Engineering: Compute moments of inertia for objects with complex shapes.
  • Real-World Use: In machine learning, double integrals help in probability density functions for 2D data distributions.

These examples show how double integrals go beyond theory and are used in practical scenarios.

6. Summary Table

For clarity, here’s a table summarizing the key aspects of double integrals and calculators:

Aspect Description Key Points
Definition Integral over a 2D region to find volumes, areas, etc. Formula: $$ \iint_R f(x,y) , dA $$
Computation Steps Set limits, integrate inner then outer variable. Order matters; use calculators for complex cases.
Common Tools Wolfram Alpha, Symbolab, Desmos. Free, step-by-step, and graphical support.
Applications Volume, center of mass, probability. Used in physics, engineering, and data science.
Challenges Irregular regions may require coordinate changes. Calculators handle this efficiently.
Result Accuracy High with symbolic tools; approximate with numerical methods. Always verify with context.

7. Final Summary

In summary, a double integral calculator is an essential tool for computing double integrals efficiently, especially when dealing with complex functions or regions. We covered the basics: double integrals extend single integrals to two dimensions, and they can be computed step by step or using online tools like Wolfram Alpha and Symbolab. I demonstrated a simple computation and provided examples to show real-world applications. Remember, understanding the process manually builds intuition, while calculators save time for advanced problems.

If the image generation function call succeeds, it will provide a visual aid to further enhance your learning. If you have a specific double integral or need more details, feel free to ask!

@Dersnotu

What is a double integral calculator and how can it be used?

Answer: A double integral calculator is a computational tool or software that helps evaluate double integrals, which are used in multivariable calculus to find volumes under surfaces, areas, or other quantities over a two-dimensional region. Double integrals extend the concept of single integrals by integrating a function of two variables, such as f(x, y), over a region in the xy-plane. This can be done manually using calculus techniques or with digital tools like online calculators, graphing software (e.g., Wolfram Alpha, MATLAB, or Python with libraries like SymPy), which automate the process and provide visual representations.

In your query, you mentioned “doubleintegral calculator,” which I interpret as a request for information or assistance with double integrals, possibly including how to use such a calculator or compute examples. I’ll provide a detailed explanation, step-by-step examples, and guidance on using calculators, while incorporating visual aids through MathJax for clarity. Since the function call to generate an image failed earlier, I’ll describe the concepts visually using text and equations instead, and suggest free tools you can use.

Table of Contents

  1. Overview of Double Integrals
  2. Key Terminology
  3. How Double Integrals Work
  4. Step-by-Step Calculation of a Double Integral
  5. Using a Double Integral Calculator
  6. Common Applications and Examples
  7. Comparison of Double Integral Tools
  8. Potential Errors and Troubleshooting
  9. Summary and Key Takeaways

1. Overview of Double Integrals

Double integrals are a fundamental concept in calculus for handling functions of two variables. They calculate the accumulated quantity (like volume or mass) over a two-dimensional area. For instance, if you have a function f(x, y) that represents height or density over a region, a double integral computes the volume under that surface.

The general form of a double integral is:

\iint_R f(x, y) \, dA

where:

  • R is the region of integration in the xy-plane.
  • dA represents the area element, often written as dx dy or dy dx depending on the order of integration.

Double integrals can be evaluated in two ways:

  • Iterated integrals: Compute the integral step by step, first with respect to one variable, then the other.
  • Using calculators or software: For complex regions or functions, tools like online calculators save time by handling the computations and providing graphs.

These integrals are widely used in fields like physics (e.g., finding mass distributions), engineering (e.g., calculating moments of inertia), and economics (e.g., optimizing resource allocation over a region).

2. Key Terminology

Before diving deeper, let’s define some key terms to ensure clarity:

  • Double Integral: An integral over a two-dimensional region, used to compute quantities like area, volume, or average value of a function.
  • Region of Integration (R): The area in the xy-plane over which the integral is computed, which can be rectangular, circular, or irregular.
  • Iterated Integral: A double integral broken into two single integrals, e.g., $$ \int_a^b \int_c^d f(x, y) , dy , dx $$.
  • Fubini’s Theorem: Allows changing the order of integration (dx dy vs. dy dx) if the function is continuous, which simplifies calculations.
  • Volume Under a Surface: The physical interpretation of a double integral where f(x, y) ≥ 0; it gives the volume between the surface and the xy-plane.
  • Calculator/Software: Digital tools that automate integration, often providing numerical approximations, exact solutions, and 3D visualizations.

3. How Double Integrals Work

A double integral works by summing infinitesimal contributions over a 2D region. If you imagine a surface defined by f(x, y), the double integral adds up the “heights” of the function across the area to find a total quantity.

For a rectangular region, the process is straightforward:

  • Define the limits for x and y.
  • Integrate with respect to one variable first, treating the other as constant.
  • Then integrate the result with respect to the second variable.

For non-rectangular regions, you may need to use coordinate transformations (e.g., polar coordinates) or split the region into simpler parts.

Visual Description: If we were to plot a function like f(x, y) = x + y over a rectangle from x=0 to 1 and y=0 to 1, it would show a plane tilting upward. The double integral would compute the volume under this plane, forming a pyramid-like shape. (Unfortunately, the image generation attempt failed, but you can visualize this using free tools like GeoGebra or Desmos.)

4. Step-by-Step Calculation of a Double Integral

Let’s solve a simple example step by step: Compute the double integral of f(x, y) = x + y over the rectangular region R where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.

Step 1: Set Up the Integral

The double integral is:

\iint_R (x + y) \, dA = \int_0^1 \int_0^1 (x + y) \, dy \, dx

We’ll integrate with respect to y first, then x.

Step 2: Integrate with Respect to the First Variable (y)

Treat x as a constant and integrate f(x, y) = x + y with respect to y:

\int_0^1 (x + y) \, dy = \left[ x y + \frac{y^2}{2} \right]_0^1

Evaluate at the limits:

  • At y = 1: x \cdot 1 + \frac{1^2}{2} = x + \frac{1}{2}
  • At y = 0: x \cdot 0 + \frac{0^2}{2} = 0

So, the result is:

x + \frac{1}{2} - 0 = x + \frac{1}{2}

Step 3: Integrate the Result with Respect to the Second Variable (x)

Now integrate the expression with respect to x from 0 to 1:

\int_0^1 \left( x + \frac{1}{2} \right) \, dx = \left[ \frac{x^2}{2} + \frac{1}{2}x \right]_0^1

Evaluate at the limits:

  • At x = 1: \frac{1^2}{2} + \frac{1}{2} \cdot 1 = \frac{1}{2} + \frac{1}{2} = 1
  • At x = 0: \frac{0^2}{2} + \frac{1}{2} \cdot 0 = 0

So, the result is:

1 - 0 = 1

Step 4: Interpret the Result

The double integral equals 1, which represents the volume under the surface f(x, y) = x + y over the unit square. This can also be verified using Fubini’s Theorem by switching the order of integration (integrating dx first, then dy), which should give the same result.

Bold Key Result: The value of the double integral is 1.

5. Using a Double Integral Calculator

Double integral calculators are user-friendly tools that handle complex computations quickly. Here’s how to use one:

How to Use an Online Calculator

  1. Choose a Tool: Popular free options include:
  • Wolfram Alpha: Enter “double integral of (x+y) dx dy from x=0 to 1, y=0 to 1” for instant results and graphs.
  • Symbolab: Type the function and limits; it provides step-by-step solutions.
  • Desmos or GeoGebra: Great for visualizing the surface and volume; input the function and set the region.
  1. Input the Function and Limits:
  • Specify the function, e.g., f(x, y) = x + y.
  • Define the region, e.g., x from 0 to 1, y from 0 to 1.
  • Choose the order of integration if needed.

  1. Run the Calculation: The tool will output the numerical or exact value, often with a 3D plot.

  2. Example in Wolfram Alpha:

  • Query: “Evaluate double integral of x+y over x=0 to 1, y=0 to 1”
  • Output: Shows the result (1 in this case) and a 3D graph.

Advantages of Calculators

  • Speed: Automates tedious calculations.
  • Visualization: Provides 3D graphs to help understand the geometry.
  • Error Reduction: Minimizes mistakes in manual integration.

If you’re using programming, here’s a quick Python example with SymPy:

from sympy import symbols, integrate

x, y = symbols('x y')
f = x + y # Define the function

# Integrate with respect to y first, then x
result = integrate(f, (y, 0, 1), (x, 0, 1))
print(result) # Output: 1

This code gives the exact value and can be extended for more complex regions.

6. Common Applications and Examples

Double integrals have practical uses across disciplines. Here are some examples:

  • Volume Calculation: Find the volume under a surface, e.g., f(x, y) = x^2 + y^2 over a disk.
  • Area and Mass: Compute the area of a region or mass distribution with density functions.
  • Probability: In statistics, double integrals find joint probabilities for two random variables.

Example 2: Double Integral in Polar Coordinates
Compute the double integral of f(x, y) = x^2 + y^2 over a circular region of radius 1 centered at the origin. Use polar coordinates (r, θ) where x = r cos θ, y = r sin θ, and dA = r dr dθ.

Step-by-Step:

  1. Change to polar: f(x, y) = r^2, limits: r from 0 to 1, θ from 0 to 2π.
  2. Integral: $$ \int_0^{2\pi} \int_0^1 r^2 \cdot r , dr , d\theta = \int_0^{2\pi} \int_0^1 r^3 , dr , d\theta $$
  3. Integrate r: $$ \int_0^1 r^3 , dr = \left[ \frac{r^4}{4} \right]_0^1 = \frac{1}{4} $$
  4. Integrate θ: $$ \int_0^{2\pi} \frac{1}{4} , d\theta = \frac{1}{4} \cdot 2\pi = \frac{\pi}{2} $$

Bold Key Result: The volume is π/2.

7. Comparison of Double Integral Tools

Here’s a table comparing popular double integral calculators and software:

Tool Ease of Use Features Cost Best For
Wolfram Alpha High (web-based) Step-by-step solutions, 3D graphs, exact and numerical results Free (basic), paid for advanced Quick checks and visualizations
Symbolab High (app/web) Detailed steps, supports various regions, educational focus Free (with ads), premium for no ads Students learning calculus
Desmos Medium (interactive) 3D graphing, customizable, shareable links Free Visual learners and exploration
Python (SymPy) Low to medium (coding required) Programmable, handles complex regions, exportable results Free (open-source) Advanced users and automation
MATLAB Medium Numerical integration, 3D plotting, supports large datasets Paid (academic discounts) Engineers and researchers

8. Potential Errors and Troubleshooting

When using double integral calculators or computing manually, common issues include:

  • Incorrect Limits: Ensure the region R is defined properly; mismatched limits can lead to errors.
  • Order of Integration: If the function is not symmetric, switching dx and dy might cause issues—use Fubini’s Theorem to check.
  • Non-Rectangular Regions: For circles or other shapes, use coordinate transformations to simplify.
  • API Errors: As seen in the failed image generation attempt, tools might require specific parameters (e.g., size for images). If using software, check documentation for required inputs.

Troubleshooting Tips:

  • Verify your function and limits.
  • Use online forums or documentation for help.
  • For visualization errors, try alternative tools like GeoGebra.

9. Summary and Key Takeaways

A double integral calculator is an essential tool for evaluating integrals over 2D regions, providing both computational results and visual insights. In the example of f(x, y) = x + y over [0,1] × [0,1], the integral equals 1, representing the volume under the surface. Double integrals are crucial for applications like volume and mass calculations, and tools like Wolfram Alpha or Python can simplify the process.

Key Takeaways:

  • Always define the region and function clearly.
  • Use iterated integrals for manual computation and calculators for efficiency.
  • For visualization, explore free tools like Desmos to see 3D representations.
  • If you need help with a specific double integral or calculator, provide more details for tailored assistance.

If you’d like to compute a different example or need help with a specific function, let me know! @Dersnotu