Question

Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.

${\int }_0^2{\int }_1^4{\int }_{-2}^{3}dxdydz$

Short answer

The value of the triple integral is 30.

Step-by-step solution

Step 1. Given information.

Consider the given information.

${\int }_0^2{\int }_1^4{\int }_{-2}^{3}dxdydz$

Step 2. Graph the solid.

By using the limits, sketch the given solid.

$\begin{array}{l}-2\le x\le 3\\ 1\le y\le 4\\ 0\le z\le 2\end{array}$

And,

Step 3. Calculate the integral.

Integrate by using limit and solve.

$\begin{array}{rcl}{\int }_0^2{\int }_1^4{\int }_{-2}^{3}dxdydz& =& {\int }_0^2{\int }_1^4{\left[x\right]}_{-2}^{3}dydz\\ & =& {\int }_0^2{\int }_1^4\left[3-\left(-2\right)\right]dydz\\ & =& 5{\int }_0^2{\left[y\right]}_1^{4}dz\\ & =& 5{\int }_0^2\left[4-1\right]dz\\ & =& 15{\left[z\right]}_0^2\\ & =& 30\end{array}$