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Chapter 13: Q. 54 (page 1016)

In Exercises 49鈥�58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
${鈭�}_{0}^{1} {鈭�}_{- e^{x}}^{e^{x}} \sin (e^{x}) d y d x$

Short Answer

Expert verified

Value of the given integral is: $2 [- \cos (e) + \cos (1)]$

Step by step solution

01

Step 1. Given Information

An integral, ${鈭�}_{0}^{1} {鈭�}_{- e^{x}}^{e^{x}} \sin (e^{x}) d y d x$

02

Step 2. Sketching the region of integration

Region of integration is shaded in the graph below.

03

Step 3. Evaluating the given iterated integral

Given integral,

{鈭�}_{0}^{1} {鈭�}_{- e^{x}}^{e^{x}} \sin (e^{x}) d y d x = {鈭�}_{0}^{1} \sin (e^{x}) {(y)}_{- e^{x}}^{e^{x}} d x = {鈭�}_{0}^{1} \sin (e^{x}) (e^{x} - (- e^{x})) d x = {鈭�}_{0}^{1} 2 e^{x} \sin (e^{x}) d x

Put, $e^{x} = t, e^{x} d x = d t,$

We get, ${鈭�}_{1}^{e} 2 \sin (t) d t$

$= 2 {[- \cos (t)]}_{1}^{e} = 2 [- \cos (e) + \cos (1)]$

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Most popular questions from this chapter

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.

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral ${鈭�}_{a}^{b} {鈭�}_{c}^{d} f (x, y) d y d x$ .

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

{鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k)

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

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.

role="math" localid="1650327788023" $鈭� {鈭�}_{R} y \sin x d A,$

whererole="math" localid="1650327080219" $R = \{(x, y) | 0 鈮� x 鈮� \frac{蟺}{2} a n d 0 鈮� y 鈮� 1\}$

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