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Chapter 13: Q. 2 (page 1082)

Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let
$R = \{(x, y) | 0 鈮� x 鈮� 2 a n d 1 鈮� y 鈮� 4\}$ .
role="math" $鈭� (x y^{2}) d A$

Short Answer

Expert verified

$鈭� (x y^{2}) d A = 42$ square units

Step by step solution

01

Definition of double integral

Let $R = \{(x, y) | a 鈮� x 鈮� b a n d c 鈮� y 鈮� d\}$ and $f (x, y)$ is continuous on R. Then.

$鈭� f (x, y) d A = {鈭�}_{a}^{b} {鈭�}_{c}^{d} f (x, y) d x d y$

02

Evaluate the integral

$鈭� (x y^{2}) d A = {鈭�}_{1}^{4} {鈭�}_{0}^{2} (x y^{2}) d x d y = {鈭�}_{1}^{4} [{鈭�}_{0}^{2} (x y^{2}) d x] d y = {鈭�}_{1}^{4} d y {[\frac{x^{2}}{2} y^{2}]}_{0}^{2} = {鈭�}_{1}^{4} d y [\frac{2^{2}}{2} y^{2}] = {鈭�}_{1}^{4} (2 y^{2}) d y = {[2 \frac{y^{3}}{3}]}_{1}^{4} = \frac{128}{3} - \frac{2}{3} = \frac{126}{3} = 42$

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{0}^{2} {鈭�}_{0}^{3} {鈭�}_{4 x / 3}^{4} f (x, y, z) d z d x d y$

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 $13.14$ .

In Exercises 61鈥�64, let R be the rectangular solid defined by $R = \{(x, y, z) | 0 鈮� a_{1} 鈮� x 鈮� a_{2}, 0 鈮� b_{1} 鈮� y 鈮� b_{2}, 0 鈮� c_{2} 鈮� z 鈮� c_{2}\} .$

Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

What is the difference between a triple integral and an iterated triple integral?

Evaluate the sums in Exercises $23 鈥� 28$ .

${鈭�}_{i = 1}^{4} {鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{2} i j^{2} k^{3}$

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