# Triple Integral Calculator

Try out our triple integral calculator with steps to get the solution of triple integral values of the functions in detail.

## Introduction to the Triple Integral Calculator

The triple integration calculator is an online calculator which is used to determine the triple integrand values of the respective function. It integrates the values of triple integrand through step-by-step calculations.

It is convenient and easy to use tool that gives you step-by-step instructions to solve the complicated problems of calculus. The best attribute of the triple integral solver is that it is free of cost, understandable, and gives you authentic results.

Along with this super tool, you can also use area under the graph calculator for the calculation of exact area under the curve online.

## What is the Triple Integration Calculator?

The triple integral calculator with steps means that it calculates the integrands in a three-dimensional direction. It evaluates the integrals of 3D shaped body. It calculates the volume, area, and mass of anybody whose density is variable and known.

The function which is to be calculated is expressed as:

F(x,y,z)

The order of integration on this integrand is x, y, and then z. The triple integration calculator also shows you results in the form of curves, graphs, and plots.

If function is the form of f(x,y), then using the double integrals calculator is the best option.

## How does Triple Integral Solver Work?

This 3d integral calculator finds the limit of the sum of the product of a function by following these steps:

1. First of all, enter a function with respect to x, y, and z variables whose triple integrals you want to calculate.
2. Choose the respective order of variables for it to differentiate e.g dxdydz, dydxdz or dzdxdy respectively.
3. Select the upper and lower limits for the variables.
4. Now simply hit, the "Calculate Triple Integral" button to get the results quickly.

The volume integral calculator displays the results on your screen, and the results are shown with step-wise calculations.

Right after that, if you want to know either the function is convergent or divergent then use of integral convergence calculator is quite appropriate.

## Formula Used By Triple Integral Calculator

The 3d integral calculator for solving the 3D integrals uses the following formula to get the accurate volume of three dimensional body/area. This formula is given as:

$$\int_{z_1}^{z_2} \left( \int_{y_1}^{y_2} \left( \int_{x_1}^{x_2} f(x,y,z)dx \right) dy \right) dz$$

Here, f(x,y,z) is a 3d integrand depending upon three different variable.

Let’s learn how this triple integrals calculator uses the above mentioned formula to solve any integral

Example: A cube has sides of length 2. Let one corner be at the origin and the adjacent corners be on the positive x, y, and z axis.

If the cube's density is proportional to the distance from the xy-plane, find its mass.

Solution: The density of the cube is f (x, y, z) = kz for some constant k.

If W is the cube, the mass is the triple integral

$$\int \int \int W \;kz\; dV \;=\; \int_0^2 \int_0^2 \int_0^2 kz \; dx dy dz$$ $$\int_0^2 \int_0^2 (x = 2x = 0)dy dz$$ $$\int_0^2 \int_0^2 2kz \; dy dz$$ $$\int_0^2 (y = 2y = 0)dy dz$$ $$\int_0^2 4k dz \;=\; 4kdz^2 | z \;=\; 2z \;=\; 0 \;=\; 16k$$

## Attributes of Triple Integral Calculator with Steps

The triple integral solver is the integral online calculator used to solve the triple integrands of the given function with respect to different variables. It evaluates the volume and mass of the given function with respect to three different variables.

The triple integrals calculator gives you authentic results in no time. It is a free of cost tool that doesn’t charge you for any subscription payments before and after its use.

This triple integral calculator calculates the integrals in three dimensions. It also shows the results in the form of graphs, curves, and plots. It gives you reliable and easily understandable results. By using triple integration calculator, one should not worry about the accuracy and authentication of the complex problems of calculus.

### Evaluate the Triple Integral Where is the Solid Tetrahedon with Vertices .

Triple integration helps to determine the volume and find the functions in three-dimensional regions which you can also determine using the triple integral volume calculator.

To calculate the triple integral over a solid tetrahedron, integrate over a range defined by the geometric limit of the solid. Here are some steps to find the triple integral over the solid tetrahedron,

• Calculate the vertices of the tetrahedron and derive the limits for each variable. This involves setting the planes and intersecting to create boundary conditions.
• Now write the triple integral in terms of the bounds and integrate with respect to each variable in order.

### Evaluate the Triple Integral. e y dv, Where e = (x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ x, x − y ≤ z ≤ x + y

To calculate the triple integral use the following,

$$\int_{x=0}^{4} \int_{y=0}^{x} \int_{z=x-y}^{x+y} y\;dz\;dy\;dx$$

First of all, integrate y with respect to z, we have:

$$\int_{x=0}^{4} \int_{y=0}^{x} y . (x + y - (x - y)) dy\;dx$$

Expand the integral,

$$\int_{x=0}^{4} \int_{y=0}^{x} 2y\;dy\;dx$$

Integrate 2y with respect to y,

$$\int_{x=0}^{4} [y^2]_{y=0}^{x} dx \;=\; \int{x=0}^{4} x^2 dx$$

Now integrate x2 with respect to x,

$$\frac{x^3}{3} \biggr|_{x=0}^{4} \;=\; \frac{4^3}{3} \;=\; \frac{64}{3}$$

So the final result for the given triple integral is,

$$\frac{64}{3} \approx 21.33$$

### What is a Triple Integral

The triple integral is the extension of a single integral and double integral into three-dimensional space. It is used to integrate the function over a three-dimensional region and is commonly applied to determine the volumes, centroids, masses, and three-dimensional properties. The general form of triple integrals are given by triple integral grapher is,

$$\int_{a}^{b} \int_{c(x)}^{d(x)} \int_{e(x,y)}^{x,y} f(x,y,z) dz \;dy\;dx$$

Triple integrals are used to determine the volume of the three-dimensional region. It also helps to find the mass when integrated with a density of the function which can calculate the mass of a 3D object. Triple integrals help in finding the centroids, moments of inertia, and centers of mass.

### How to Convert x y z to Cylindrical Coordinates Triple Integral

During the calculation in the integral triple calculator, Converting the triple integral from cartesian coordinates to cylindrical coordinates involves changing the variable representation and putting the Jacobian determinant for the transformations’ effect on the differential volume element.

Cartesian to Cylindrical Coordinates:

In cylindrical coordinates,

$$x \;=\; r cos(\theta)$$

$$y \;=\; r sin(\theta)$$

$$z \;=\; z$$

The differential volume in the cylindrical coordinates is.

$$dV \;=\; r\;dr\;d \theta dz$$

• Rewrite the function which is to be integrated using the cylindrical coordinates

$$x \;=\; r\;cos(\theta), y \;=\; r\;sin\; (\theta)\;and\; z \;=\; z$$

• Adjust the differential volume element including the Jacobian determinant in the differential volume element

$$dV \;=\; r\;dr\;d \theta dz$$

• Now determine the limits of integrations depending on the geometry. Set the limits to cover the desired radical range and establish the angular limits to define the vertical limits.

### Evaluate the Triple Integral dv Where e Lies Between the Spheres x^2+y^2+z^2=4 and x^2+y^2+z^2=9

To calculate the triple integral in which the volume is between the spheres x2 + y2 + z2= 4 and x2 + y2 + z2= 9, the conversion to cartesian coordinates is given by,

$$x \;=\; r\; sin \varnothing cos \theta$$

$$y \;=\; r\; sin \varnothing sin \theta$$

$$z \;=\; r\; cos \varnothing$$

Therefore, the differential volume is given by,

$$dv \;=\; r^2 sin \varnothing dr\; d \varnothing d \theta$$

For the integration limits,

The radical distance ranges from the inner sphere to the outer sphere and the polar angle ranges from 0 to π.

To find the volume of the region, the triple integral between the two spheres is,

$$\int_{0}^{2 \pi} \int_{0}^{\pi} \int_{2}^{3} r^2 sin \varnothing dr\; d \varnothing d \theta$$

Integrating with respect to r,

$$\int_{2}^{3} r^2 dr \;=\; \frac{1}{3} r^3 \biggr|_{2}^{3} \;=\; \frac{1}{3} (27 - 8) \;=\; \frac{19}{3}$$

Now,

$$\int_{0}^{\pi} sin \varnothing d \varnothing \;=\; -cos \varnothing \biggr|_{0}^{\pi} \;=\; 2$$

$$\int_{0}^{2 \pi} d \theta \;=\; 2 \pi$$

Combining the results,

$$\int_{0}^{2 \pi} \int_{0}^{\pi} \int_{2}^{3} r^2 sin \varnothing dr\; d \varnothing d \theta \;=\; \frac{19}{3} \times 2 \times 2 \theta \;=\; \frac{38 \pi}{3}$$