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Chapter 13: Q. 53 (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.
${鈭�}_{0}^{\frac{蟺}{4}} {鈭�}_{t a n x}^{s e c x} y d y d x$

Short Answer

Expert verified

Value of the given integral is: $\frac{蟺}{8}$

Step by step solution

01

Step 1. Given Information

An integral, ${鈭�}_{0}^{\frac{蟺}{4}} {鈭�}_{t a n x}^{s e c x} y d y d x$

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}^{\frac{蟺}{4}} {鈭�}_{t a n x}^{s e c x} y d y d x = {鈭�}_{0}^{\frac{蟺}{4}} {(\frac{y^{2}}{2})}_{\tan x}^{s e c x} d x = {鈭�}_{0}^{\frac{蟺}{4}} (\frac{s e c^{2} x - \tan^{2} x}{2}) d x = {鈭�}_{0}^{\frac{蟺}{4}} (\frac{1}{2}) d x = {(\frac{x}{2})}_{0}^{\frac{蟺}{4}} = \frac{\frac{蟺}{4} - 0}{2} = \frac{蟺}{8}

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Most popular questions from this chapter

Let $f (x)$ be an integrable function on the rectangular solid $R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\}$ , and let $魏鈭� 鈩� .$ Use the definition of the triple integral to prove that:

${鈭�}_{R} 魏 f (x, y, z) d V = 魏 {鈭�}_{R} f (x, y, z) d V .$

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

$鈭� {鈭�}_{R} x y^{3} d A W h e r e R = {(x, y) | - 2 鈮� x 鈮� 2, - 1 鈮� y 鈮� 1}$

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

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} {鈭�}_{0}^{9 - y^{2} - z^{2}} f (x, y, z) d x d z d y$

Use the results of Exercises 59 and 60 to find the centers of masses of the lamin忙 in Exercises 61鈥�67.

In the following lamina, all angles are right angles and the density is constant:

$Express the sum 3 e^{4} + 3 e^{9} + 3 e^{16} + 4 e^{4} + 4 e^{9} + 4 e^{16} using double - summation notation .$

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