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Triple Integral Calculator Level 2

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Description

Triple Integral Calculator Level 2 calculates definite triple integrals of real functions with three real variables. Numerical values are calculated with precision up to 15 digits. The calculator does not make any symbolic integration and any analytical guessing. The algorithm is purely numerical. All calculation results are reflected in History rich-text-box, which can be saved in file or printed.

How to Use the Triple Integral Calculator?

The procedure to use the triple integral calculator is as follows:
Step 1: Enter the functions and limits in the respective input field
Step 2: Now click the button “Calculate” to get the integrated value
Step 3: Finally, the integrated value will be displayed in the new window

What is Meant by the Triple Integrals?

In Maths, the triple integral is similar to the double integral as a limit of Riemann sum. Generally, triple integrals are used for integrating over the three-dimensional space. Triple integrals are used to find the volume like the double integrals. But it also calculates the mass, when the volume of the region has the variable density. The function can be represented by f(x, y, z).

What is Triple Integral?

In mathematics, the triple integral is same as the single or double integral. Normally, triple integration is used to integrating over the three-dimensional space. Triple integral used to determine the volume like the double integrals. But it also determines the mass, when the volume of any body has variable density. The function can be expressed as:

What does a Triple Integral Actually Calculate?

The above calculator finds the solution to your definite triple integral within the x, y, and z integration limits that you inputted. Calculating a triple integral allows us to find the volume contained within of a 3-d shape. The difference between a triple and a double integral is that double integrals find the volume under a surface, whereas a triple integral finds the volume between two continuous surfaces that comprise a shape.

One of the most difficult parts of calculating triple integrals is visualizing it and then setting up the limits of integration. To understand how to set up and calculate the integral, it is important to understand how calculus operations are able to calculate volume of a 3D shape. In the next section we will explain the process.

In triple integral, you need to evaluate the integration for three variables with respect to three different variables. You have to study the following module to get the ideas on how to evaluate the triple integral. Follow these steps and determine the functions with triple integral calculator or manually.

Take a function that have three different variables to figure out the triple integral.
Firstly, perform the integration with one variable to eliminate the certain variable.
Now, substitute the obtained values in the expression like upper and lower limit.
While performing integration on one variable, you have to consider the other variables as constants.
After eliminating the one variable, you should repeat the process to eliminate the other variables to obtain the answer in constant.

However, an online Integral Calculator allows you to evaluate the integrals of the functions with respect to the variable involved.

Example:

Question: Solve $\int_{2}^{3} \int_{1}^{3} \int_{0}^{1} (x^{2} + 3 x y z^{2} + x y z)$ dxdydz?

Solution:

First, take the inner integral

$\int (x^{2} + 3 x y z^{2} + x y z) d x$

Integrate term-by-term:

The integral of $x^{n} i s x^{n} + 1 / n + 1$ when n ≠ −1:

$\int x^{2} d x = x^{3} / 3$

$\int 3 x y z^{2} d x = 3 y z^{2} \int x d x$

The integral of $x^{n} i s x^{n + 1} / n + 1$ when n ≠ −1:

$\int x d x = x^{2} / 2$

So, the result is: $3 x^{2} y z^{2} / 2$

$\int x y z d x = y z \int x d x$

The integral of $x^{n} i s x^{n + 1} / n + 1$ when n ≠ −1:

$\int x d x = x^{2} / 2$

So, the result is: $x^{2} y z / 2$

The result is: $x^{3} / 3 + 3 x^{2} y z^{2} / 2 + x^{2} y z / 2$

Now, triple integral calculator simplifies the obtain values:

$X^{2} (2 x + 9 y z^{2} + 3 y z) / 6$

Add the constant of integration:

$X^{2} (2 x + 9 y z^{2} + 3 y z) / 6 + c o n s t a n t$

The answer is:

$X^{2} (2 x + 9 y z^{2} + 3 y z) / 6 + c o n s t a n t$

Then we take second integral:

$\int x^{2} (x / 3 + y z (3 z + 1) / 2) d y$

$\int x^{2} (x / 3 + y z (3 z + 1) / 2) d y = x^{2} \int (x / 3 + y z (3 z + 1) / 2) d y$

Integrate term-by-term:

The integral of a constant is the constant times the variable of integration:

$\int x / 3 d y = x y / 3$

$\int y z (3 z + 1) / 2 d y = z (3 z + 1) \int y d y / 2$

The integral of $y^{n} i s y^{n + 1} / n + 1$ when n ≠ −1:

$\int y d y = y^{2} / 2$

So, the result is: $y^{2} z (3 z + 1) / 4$

$X y / 3 + y^{2} z (3 z + 1) / 4$

So, the result is: $x^{2} (x y / 3 + y^{2} z (3 z + 1) / 4)$

Now, the tripple integral calculator simplify:

$X^{2} y (4 x + 3 y z (3 z + 1)) / 12$

Add the constant of integration:

$X^{2} y (4 x + 3 y z (3 z + 1)) / 12 + c o n s t a n t$

The answer is:

$X^{2} y (4 x + 3 y z (3 z + 1)) / 12 + c o n s t a n t$

At the end, triple integral solver take third integral:

$\int x^{2} y (4 x + 3 y z (3 z + 1)) / 12 d z$

$\int x^{2} y (4 x + 3 y z (3 z + 1)) / 12 d z = x^{2} y \int (4 x + 3 y z (3 z + 1)) d z / 12$

Integrate term-by-term:

$\int 4 x d z = 4 x z$

The integral of a constant times a function is the constant times the integral of the function:

$\int 3 y z (3 z + 1) d z = 3 y \int z (3 z + 1) d z$

Rewrite the integrand:

$Z (3 z + 1) = 3 z^{2} + z$

Now, triple integral calculator integrates term-by-term:

$\int 3 z^{2} d z = 3 \int z^{2} d z$

The integral of $z^{n} i s z^{n + 1} / n + 1$ when n ≠ −1:

$\int z^{2} d z = z^{3} / 3$

So, the result is: $z^{3}$

The integral of $z^{n} i s z^{n + 1} / n + 1$ when n ≠ −1:

$\int z d z = z^{2} / 2$

The result is: $z^{3} + z^{2} / 2$

$3 y (z^{3} + z^{2} / 2)$

$4 x z + 3 y (z^{3} + z^{2} / 2)$

So, the result is: $x^{2} y (4 x z + 3 y (z^{3} + z^{2} / 2)) / 12$

Now, triple iterated integral calculator simplify the obtaining values:

$X^{2} y z (8 x + 3 y z (2 z + 1)) / 24$

Then, triple integration calculator adds the constant of integration:

$X^{2} y z (8 x + 3 y z (2 z + 1)) / 24 + c o n s t a n t$

The answer is:

$X^{2} y z (8 x + 3 y z (2 z + 1)) / 24 + c o n s t a n t$

Integration in Cylindrical Coordinates:

Triple integrals are usually calculated by using cylindrical coordinates than rectangular coordinates. Some equations in rectangular coordinates along with related equations in cylindrical coordinates are listed in Table. The equations become easy as cylindrical integral calculator proceed with solving problems using triple integrals.

	Circular Cylinder	Circular cone	Sphere	Paraboloid
Cylindrical	R = c	Z = cr	$R^{2} + z^{2} = c^{2}$	$Z = c r^{2}$
Rectangular	$X^{2} + y^{2} = c^{2}$	$Z^{2} = c^{2} (x^{2} + y^{2})$	$X^{2} + y^{2} + z^{2} = c^{2}$	$Z = c (x^{2} + y^{2})$

However, an online Derivative Calculator helps to determine the derivative of the function with respect to a given variable.

How does Triple Integral Calculator Works?

An online triple integrals calculator can find the limit of the sum of the product of a function by follow these steps:

Input:

First, enter a function with respect to x, y, and z variables.
If you know the upper and lower limit for variables, then choose the definite and substitute upper and lower limits.
Apart from this, if you have no idea about the limits of variables, then select the indefinite.
Hit the Calculate Triple Integral button.

Output:

The cylindrical integral calculator displays the indefinite and definite integral with step-wise calculations.

FAQ:

Why triple integral can be used?

The triple integral mostly used to determine the mass and volume just like the double integral.

What is the volume integral used for?

In calculus, a volume integral refers to the integral over a three-dimensional domain. It is a special case of multiple integrals. In order to calculate flux densities volume integral most commonly used in physics.

How to find the volume of triple integral?

The ellipsoid volume can be represented as the triple integral that is $V = ∭_{U} d x d y d z = ∭_{U}^{'} a b c p^{2} s i n θ d p d φ d θ$ . By symmetry, you can evaluate the volume of ellipsoid lying in the first octant and multiply the results by 8.

Conclusion:

Use this online triple integral calculator to determine the triple integral of entered functions. Triple integral mostly used to determine the volume and mass for the functions with three different variables to integrate over the given interval.

Reference:

From the source of Wikipedia: Multiple integral, Riemann integrable, Methods of integration, integrating constant functions, Use of symmetry, Normal domains on R2.
From the source of Libre Text: Triple Integrals in Cylindrical Coordinates, Integration in Cylindrical Coordinates, Fubini’s Theorem in Cylindrical Coordinates.
From the source of Lumen Learning: Double Integrals Over Rectangles, Double Integrals Over Rectangles, Iterated Integrals, Double Integrals Over General Regions.

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How to Solve Triple Integrals

We perform triple integrals on functions of three variables. Just like a double integral, we solve triple integrals by iterated integrals. Pictured below is the notation of a definite triple integral.

As we can see, the triple integral ends up being three iterated integrals in which the order is determined by how the notation is set up. The iterated integrals are performed from innermost to outermost. In this case, the order goes dx, then dy, then dz. The limits of integration for each variable are ordered to match dx, dy, and dz.

The limits of integration are written as functions of the limits to their outside because of the iterated nature of a triple integral. Using the image above as an example, dx is integrated first so the result of that remains a function of y and z. After dy, it remains a function of z.

How the Triple Integral Calculator Works

The calculator on this page computes your triple integral symbolically by using a computer algebra system. In symbolic integration, the computer algebra system uses algebra and integral rules to take the antiderivative of the function before applying the fundamental theorem of calculus. In fact, it iterates through the integral just as described in the section above. In essence, symbolic integration follows the same steps as a human with a paper and pencil would. It has the capability to attain near perfect solution accuracy. The calculator on this page is accurate to a minimum of the 5th decimal place!

The alternative to using symbolic integration to solve triple integrals is called numerical integration. A numerical integration routine performs a relatively small and approximated version of the problem as many times as necessary to converge to an accurate solution. Generally, numerical routines can solve a greater range of problems but can take longer and potentially be less accurate.

Triple Integral Calculator Level 2Overview

This calculator computes definite triple integrals of real functions with three real variables,utilizing triple tinh-sinh quadrature scheme. Although some precaution about theoretical foundation of tanh-sinh quadrature exists, in practice the scheme is almost universal. Numerical values are calculated with Precision 15-16 Digits. All calculation results are reflected in History rich-text-box, which can be saved in file or printed

FAQ:

Why triple integral can be used?

The triple integral mostly used to determine the mass and volume just like the double integral.

What is the volume integral used for?

Conclusion:

Reference:

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Triple Integral Calculator Level 2