Integrate g (x, y, z) = 8 x y z over the surface of the rectangular solid cut from the first...

Question:

Integrate {eq}g (x,\ y,\ z) = 8 x y z {/eq} over the surface of the rectangular solid cut from the first octant by the planes {eq}x = a,\ y = b {/eq}. and {eq}z = c {/eq}.

Application of Triple Integrals:

To set u p an integral we need to determine the limits for integration based on the given bounding equations. Based on the given boundaries we can easily interchange the order of integrations for instance {eq}dzdydx,\:dxdydz {/eq} .

Answer and Explanation:

From the given bounding equations the limits are the following,

{eq}0\leq x\leq a,\:0\leq y\leq b,\:0\leq z\leq c {/eq}

Thus,

{eq}\displaystyle \int \int \int g (x, y, z)dV =\int_{0}^{a}\int_{0}^{b}\int_{0}^{c} 8 x y z dzdydx {/eq}

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

{eq}\displaystyle =\int_{0}^{a}\int_{0}^{b} 8 x y \left [ \frac{1}{2}z^{2} \right ]^{c}_{0}dydx {/eq}

{eq}\displaystyle =\int_{0}^{a}\int_{0}^{b}\left ( \frac{c^2}{2} \right ) 8 x y dydx {/eq}

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

{eq}\displaystyle =\int_{0}^{a}\left ( \frac{c^2}{2} \right )\left [ \frac{1}{2}y^{2} \right ]^{b}_{0} 8 x dx {/eq}

{eq}\displaystyle =\int_{0}^{a}\left ( \frac{c^2b^2}{4} \right )8 x dx {/eq}

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

{eq}\displaystyle =\left ( \frac{c^2b^2}{4} \right )8\left [ \frac{1}{2}x^{2} \right ]^{a}_{0} {/eq}

{eq}\displaystyle =a^2 b^2 c^2 {/eq}


Learn more about this topic:

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Volumes of Shapes: Definition & Examples

from GMAT Prep: Tutoring Solution

Chapter 11 / Lesson 9
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