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Chapter 13: Q. 28 (page 1082)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
${鈭�}_{0}^{3} {鈭�}_{0}^{2 - \frac{2}{3} y} {鈭�}_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y$

Short Answer

Expert verified

${鈭�}_{0}^{3} {鈭�}_{0}^{2 - \frac{2}{3} y} {鈭�}_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y = 4$

Step by step solution

01

Given

We have to find the volume by using the triple integral.

{鈭�}_{0}^{3} {鈭�}_{0}^{2 - \frac{2}{3} y} {鈭�}_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y

02

Evaluate the integral

$V = {鈭�}_{0}^{3} {鈭�}_{0}^{2 - \frac{2}{3} y} {鈭�}_{0}^{4 - \frac{4}{3} y - 2 z} d x d z d y V = {鈭�}_{0}^{3} [{鈭�}_{0}^{2 - \frac{2}{3} y} (4 - \frac{4}{3} y - 2 z) d z] d y V = {鈭�}_{0}^{3} {[4 z - \frac{4}{3} y z - z^{2}]}_{0}^{2 - \frac{2}{3} y} d y V = {鈭�}_{0}^{3} [4 (2 - \frac{2}{3} y) - \frac{4}{3} y (2 - \frac{2}{3} y) - {(2 - \frac{2}{3} y)}^{2}] d y V = {鈭�}_{0}^{3} (4 - \frac{8}{3} y + \frac{4}{9} y^{2}) d y V = {[4 y - \frac{8}{3} \frac{y^{2}}{2} + \frac{4}{9} \frac{y^{3}}{3}]}_{0}^{3} V = [12 - 12 + 4] V = 4 cubic units$

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Most popular questions from this chapter

Use Definition $13.4$ to evaluate the double integrals in Exercises $29 鈥� 32$ .

localid="1649936867482" ${鈭�}_{x} x^{2} y d A$

where $R = {(x, y) 鈭� 1 鈮� x 鈮� 3 and 0 鈮� y 鈮� 2}$

Evaluate the triple integrals over the specified rectangular solid region.

$\begin{array}{r} {鈭�}_{R} 鈥� x^{2} y z d V, where \\ R = {(x, y, z) 鈭� 鈭� 2 鈮� x 鈮� 1, 0 鈮� y 鈮� 3, and 1 鈮� z 鈮� 5} \end{array}$

Explain how to construct a midpoint Riemann sum for a function of three variables over a rectangular solid for which each $(x_{i}^{*}, y_{j}^{*}, z_{k}^{*})$ is the midpoint of the subsolid role="math" localid="1650346869585" $R_{i j k} = {(x, y, k) | x_{i - 1} 鈮� x_{i}^{*} 鈮� x_{i}, y_{j - 1} 鈮� y_{j}^{*} 鈮� y_{j} a n d z_{k - 1} 鈮� z_{k}^{*} 鈮� z_{k}}$ . Refer either to your answer to Exercise $4$ or to Definition $13.14$ .

Describe the three-dimensional region expressed in each iterated integral:

${鈭�}_{鈭� 2}^{4} 鈥� {鈭�}_{2}^{6} 鈥� {鈭�}_{0}^{5} 鈥� f (x, y, z) d y d x d z$

$How many summands are in {鈭�}_{i = 2}^{15} {鈭�}_{j = 3}^{17} {鈭�}_{k = 4}^{19} \frac{i}{j + k} ?$

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