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Triple Integral Calculator + Online Solver With Free Steps

A Triple Integral Calculator is an online tool that helps find triple integral and aids in locating a point’s position using the three-axis given:

  1. The radial distance of the point from the origin
  2. The Polar angle that is assessed from a stationary zenith direction
  3. The Point’s azimuthal angle orthogonal projection on a reference plane that passes through the origin.

It can be thought of as the polar coordinate system in three dimensions. Triple integrals over areas that are symmetrical relative to the origin can be calculated using spherical coordinates.

What Is the Triple Integral Calculator?

A Triple Integral Calculator is an online tool used to compute the triple integral of three-dimensional space and the spherical directions that determine the location of a given point in three-dimensional (3D) space depending on the distance ρ from the origin and two points $\theta$ and $\phi$. 

The calculator uses Fubini’s Theorem to evaluate triple integral because it states that if an absolute value’s integral is finite, the order of its integration is irrelevant; integrating first concerning x and then concerning y yields the same results as integrating first concerning y and then concerning x.

A triple integral function $f(\rho, \theta,\varphi)$ is formed in the spherical coordinate system. The function should be continuous and must be bounded in a spherical box of the parameters:

\[ \alpha\leq \rho \leq \beta \]

\[ \alpha \leq \theta \leq \beta \]

\[ \gamma \leq \varphi \leq \psi \]

Then each interval is divided into l, m, and n subsections.

How To Use Triple Integral Calculator?

You can use the Triple Integral calculator by specifying the values of three spherical coordinate axes. Spherical Coordinates Integral Calculator is extremely simple to use if all the necessary inputs are available.

By following the given detailed guidelines, the calculator will surely provide you with the desired results. You can therefore follow the given instructions to get the triple integral.

Step 1

Enter the triple integral function in the provided entry box and also specify the order in the adjacent box.

Step 2

Enter the upper and lower bounds of the $\rho$, $\phi$, and $\theta$ in the input field.

For $\rho$, enter the lower limit in the box named rho from and the upper limit in the box named toFor $\phi$, enter the lower limit in the box specified as phi from and the upper limit in the box specified as toFor $\theta$, enter the lower limit in theta from and the upper limit in the box named to.

Step 3

Finally, click the “Submit” button, and the whole step-by-step solution for the spherical coordinates integral will be displayed on the screen.

As we have discussed before, the calculator uses Fubini’s theorem. It has a limitation that it does not apply to the functions that are not integrable over the set of real numbers. It isn’t even bound on $\mathbb{R}$.

How Does the Triple Integral Calculator Work?

The Triple Integral Calculator works by computing the triple integral of the given function and determining the volume of the solid bounded by the function. Triple integral is exactly similar to single and double integral with the specification of integrating for three-dimensional space.

The calculator provides the step-by-step calculation of how to determine the triple integral with various methods. To further understand the working of this calculator, let’s explore some concepts related to the triple integral calculator.

What Is Triple Integral?

The Triple integral is an integral used to integrate over 3D space or to calculate the volume of a solid. The triple integral and the double integral are both limits of the Riemann sum in mathematics. Triple integrals are typically used to integrate over 3D space. The volume is determined using triple integrals, much like double integrals.

However, it also determines the mass when the region’s volume has a varied density. The function is symbolized by the representation given as:

\[f (\rho, \theta, \phi) \]

Spherical coordinates $\rho$, $\theta$, and $\phi$ are another typical set of coordinates for R3 in addition to Cartesian Coordinates given as x, y, and z. A line segment L is drawn from the origin to the point using the Spherical Coordinates Integral Calculator after selecting a location in a space other than the origin. The distance $\rho$ represents the length of line segment L, or simply, it is the separation between the origin and defined point P.

The angle between the projected line segment L and the x-axis is orthogonally projected in the x-y plane which usually fluctuates between 0 and $2\pi$. One important thing to be noted is if x, y, and z are cartesian coordinates then $\theta$ is the polar coordinate angle of the point P(x, y). The angle between the z-axis and line segment L is finally introduced as $\phi$.

The infinitesimal changes in $\rho$, $\theta$, and $\phi$ must be taken into account to get an expression for the infinite volume element dV in spherical coordinates.

How To Find the Triple Integral

The triple integral can be found by following the steps mentioned below:

  1. Consider a function with three different variables such as $ \rho $, $\phi $, and $\theta $ for calculating the triple integral for it. Triple integral requires integration with respect to three different variables.
  2. First, integrate with respect to variable $\rho$.
  3. Second, integrate with respect to the variable $\phi $.
  4. Integrate the given function with respect to $\theta $. The order of the variable matters while integrating which is why specification of the order of the variables is necessary.
  5. Finally, you will get the result after incorporating the limits.

Solved Examples

Let us solve a few examples using the Triple Integral Calculator for better understanding.

The function f(x, y, z) is said to be integrable on an interval when the triple integral occurs inside it.

Furthermore, if the function is continuous on the interval, the triple integral exists. So for our examples, we’ll consider continuous functions. Nevertheless, continuity is adequate but not mandatory; in other words, function f is constrained by the interval and continuous.

Example 1

Evaluate:

\[ \iiint_E  (16z\ dV)\] where E is the upper half of the sphere given as:

\[ x^{2} + y^{2} + z^{2} = 1\]

Solution

The variables’ limits are as follows because we are considering the upper half of the sphere:

For $\rho$:

\[ 0 \leq \ \rho\  \leq 1\]

For $\theta$:

\[0 \leq \ \theta\  \leq 2\pi \]

For $\varphi$:

\[0 \leq \ \varphi\  \leq \frac{\pi}{2}\]

The triple integral is calculated as:

\[ \int \int_{E} \int 16z \,dV = \int^{\frac{\pi}{2}}_{0} \int^{2\pi}_{0} \int^{1}_{0} \rho^2 \sin \psi (16 \rho \cos \psi) \,d\rho \,d\theta \,d \psi \]
Now, integrating with respect to $\rho$, $\theta$, and $\varphi$ respectively.
 
The equation becomes:
\[ = \int^{\frac{\pi}{2}}_{0} \int^{2\pi}_{0} \int^{1}_{0} 8\rho^3 \sin (2\psi) \,d\rho \,d\theta \,d \psi\]
\[ = \int^{\frac{\pi}{2}}_{0} \int^{2\pi}_{0} 2 \sin (2\psi) \,d\theta \,d \psi\]
\[ = \int^{\frac{\pi}{2}}_{0} 4\pi \sin (2\psi) \,d \psi\]
\[ = -2\pi \cos (2\psi) \vert ^ {\frac{\pi}{2}}\]
\[ = 4\pi\]
So, the answer is $4\pi$.

Example 2

Evaluate:

\[ \iiint_E {zx\ dV} \]

where E is inside both the function given as:

\[ x^{2} + y^{2} + z^{2} = 4\]

and the cone (pointing upward) that makes an angle of:

\[\frac{2\pi}{3}\]

with the negative z-axis and $x\leq 0$

Solution

We must first take care of the boundaries. In essence, area E is an ice cream cone that has been chopped in half, leaving just the piece with the condition:

\[ x\leq 0 \]

Consequently, since it is located inside a region of a sphere with a radius of 2, the limit must be:

\[ \ 0 \leq \rho \leq  2\]

For $ \varphi $ caution is required. The cone produces an angle of \(\frac{\pi}{3}\) with the negative z-axis, according to the statement. But keep in mind that it is calculated from the positive z-axis.

As a result, the cone will “start” at an angle of \(\frac{2\pi}{3}\), which is measured from the positive z-axis and leads to the negative z-axis. Consequently, we obtain the following limits:

\[ \frac{2\pi}{3} \leq \ \varphi\  \leq \pi\ \]

Finally, we can take the fact that x\textless0, likewise stated as evidence for the \(\theta\).

\[ \frac{\pi}{2} \leq \ \theta\  \leq \frac{3\pi}{2}\]

The triple integral is given as:

\[ \int \int_{E} \int zx \,dV = \int^{\pi}_{\frac{2\pi}{3}} \int^{\frac{3\pi}{2}}_{\frac{\pi}{2}} \int^{2}_{0} (\rho \cos \psi)(\rho \sin \psi \cos \theta)\rho^2 \sin \psi \,d\rho \,d\theta \,d \psi \]

The detailed step-by-step solution is given below:

\[ = \int^{\pi}_{\frac{2\pi}{3}} \int^{\frac{3\pi}{2}}_{\frac{\pi}{2}} \int^{2}_{0} \rho^4 \cos \psi \sin ^2 \psi \cos \theta \,d\rho \,d\theta \,d \psi\]

\[ = \int^{\pi}_{\frac{2\pi}{3}} \int^{\frac{3\pi}{2}}_{\frac{\pi}{2}} \frac{32}{5} \cos \psi \sin ^2 \psi \cos \theta  \,d\theta \,d \psi\]

\[ = \int^{\pi}_{\frac{2\pi}{3}} \frac{-64}{5} \cos \psi \sin ^ 2 \psi \,d \psi\]

\[ = – \frac{64}{15} \sin ^ 3 \psi, \frac{2\pi}{3} \leq \psi \leq \pi\]

\[ = \frac{8\sqrt{3}}{5}\]

Therefore, the Triple Integral Calculator can be used to determine the triple integral of various 3D spaces using spherical coordinates.

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