Evaluate triple integral_B f(x, y, z) dV for the specified function f and B: f(x, y, z) = xz^2...

Question:

Evaluate {eq}\displaystyle \iiint_\mathcal B f(x, y, z) dV {/eq} for the specified function {eq}f {/eq} and {eq}\mathcal B:\ f(x, y, z) = xz^2\ [0, 8] \times [7, 10] \times [3, 7] {/eq}

Evaluation of Triple Integrals:

To evaluate the given integrals the first thing we nee to do is plug -in the given limits and then we can start the integration. Note the order of integration can be interchanged for instance {eq}dzdydx,dxdzdy {/eq} and etc.

Answer and Explanation:

Plugging-in the limits we have,

{eq}\displaystyle \displaystyle \iiint_\mathcal B f(x, y, z) dV =\int_{0}^{8}\int_{7}^{10}\int_{3}^{7}xz^{2}dzdydx {/eq}

Integrate with respect to {eq}z {/eq}

{eq}\displaystyle =\int_{0}^{8}\int_{7}^{10}\left [ \frac{1}{3}z^{3} \right ]^{7}_{3}xdydx {/eq}

{eq}\displaystyle =\int_{0}^{8}\int_{7}^{10}\left ( \frac{316}{3} \right )xdydx {/eq}

Integrate with respect to {eq}y {/eq}

{eq}\displaystyle =\int_{0}^{8}\left ( \frac{316}{3} \right )\left [ y \right ]^{10}_{7}xdx {/eq}

{eq}\displaystyle =\int_{0}^{8}316xdx {/eq}

Integrate with respect to {eq}x {/eq}

{eq}\displaystyle =316\left [ \frac{1}{2}x^{2} \right ]^{8}_{0} {/eq}

{eq}=10,112 {/eq}


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Volumes of Shapes: Definition & Examples

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