# 244.T.15.5.11 Evaluate the triple integral

• MHB
• karush
In summary, The triple integral $I_{11}$ is evaluated by first evaluating the integral in the x direction, then the y direction, and finally the z direction. This results in the answer being equal to $\frac{5(2-\sqrt{3})}{4}$.
karush
Gold Member
MHB
$tiny{244.T.15.5.11}$
$\textsf{Evaluate the triple integral}\\$
\begin{align*}\displaystyle
I_{\tiny{11}}&=\int_{0}^{\pi/6}\int_{0}^{1}\int_{-2}^{3} y\sin{z}
\, d\textbf{x} \, d\textbf{y} \, d\textbf{z}\\
&=\int_{0}^{\pi/6}\int_{0}^{1}
\Biggr|xy\sin{z} \Biggr|_{-2}^{3}
\, d\textbf{y} \, d\textbf{z}\\
ans&=\color{red}{\frac{5(2-\sqrt{3})}{4}}
\end{align*}ok just want to see if these first steps are correct with xyz

I would begin by writing:

$$\displaystyle I=\int_{0}^{\pi/6}\sin(z)\int_{0}^{1}y\int_{-2}^{3}\,dx\,dy\,dz$$

As $$\displaystyle \int_{-2}^{3}\,dx=5$$ we then have:

$$\displaystyle I=5\int_{0}^{\pi/6}\sin(z)\int_{0}^{1}y\,dy\,dz$$

Now continue...:)

So then continuing..

\displaystyle \begin{align*}\displaystyle I_{11}&=\int_{0}^{\pi/6}\int_{0}^{1}\int_{-2}^{3} y\sin{z} \, dx \, dy \, dz\\ &=5\int_{0}^{\pi/6}\sin(z) \int_{0}^{1}y \,dy\,dz\\ &=5\int_{0}^{\pi/6} \sin(z)\Biggr|\frac{y^2}{2}\Biggr|_{0}^{1} \,dz\\ &=\frac{5}{2}\int_{0}^{\pi/6}\sin(z)\,dz\\ &=\frac{5}{2}\Biggr|-\cos(z)\Biggr|_{z=0}^{z=\pi/6} =\frac{5}{2}\Biggr[-\frac{\sqrt{3}}{2}-(-1)\Biggr] =\color{red}{\frac{5(2-\sqrt{3})}{4}} \end{align*}

hopefully
any suggest?

Last edited:
Incorrect notation; otherwise correct. :)

do you the vertical bars?

Ah, I see you fixed it!

Observe:

$$\left.-\frac{5}{2}\cos(z)\right|_{z=0}^{z=\pi/6}$$

Now you can use pure bars!

For something bigger:

$$\left.\left(\frac{x^2}{2}+3x+4\right)\right|^{4}_{x=0}$$

## 1. What is a triple integral?

A triple integral is a mathematical concept used in multivariable calculus to calculate the volume of a three-dimensional region.

## 2. How do I set up a triple integral?

A triple integral is typically written as ∭f(x,y,z)dV, where f(x,y,z) is the function being integrated and dV represents the infinitesimal volume element. The limits of integration for each variable (x,y,z) should be specified based on the bounds of the three-dimensional region.

## 3. How do I evaluate a triple integral?

To evaluate a triple integral, you must first set up the integral and then use appropriate integration techniques, such as the method of substitution or integration by parts, to solve for the final result. It is important to carefully consider the limits of integration and the order of integration when evaluating a triple integral.

## 4. What is the purpose of using a triple integral?

A triple integral is used to calculate the volume of a three-dimensional region, which is a useful concept in fields such as physics, engineering, and economics. It can also be used to find the mass, center of mass, and moment of inertia of a three-dimensional object.

## 5. Can a triple integral be applied to other types of regions?

Yes, a triple integral can also be used to calculate the volume of a region bounded by surfaces other than planes, such as spheres, cylinders, and cones. In these cases, the limits of integration and the equation of the infinitesimal volume element may vary.

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