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Chapter 13: Q. 22 (page 1082)

Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.
${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{4 y - y^{2}}} \frac{1}{\sqrt{x^{2} + y^{2}}} d x d y$

Short Answer

Expert verified

${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{4 y - y^{2}}} \frac{1}{\sqrt{x^{2} + y^{2}}} d x d y = 蟺$

Step by step solution

01

Draw the region

From the limits of integration, the region is shown below,

02

Convert into polar form

By using the below substitution,

$x = r \cos 胃 y = r \sin 胃 x^{2} + y^{2} = r^{2} d x d y = r d r d 胃$

The equivalent polar integral of the given integral is,

${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{4 y - y^{2}}} \frac{1}{\sqrt{x^{2} + y^{2}}} d x d y 鈬� {鈭�}_{0}^{蟺 / 2} {鈭�}_{2}^{4} d r d 胃$

03

Calculate integral

$I = {鈭�}_{0}^{蟺 / 2} {鈭�}_{2}^{4} d r d 胃 I = {[r]}_{2}^{4} {鈭�}_{0}^{蟺 / 2} d 胃 I = 2 {[胃]}_{0}^{蟺 / 2} I = 2 脳 \frac{蟺}{2} I = 蟺$

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Most popular questions from this chapter

Evaluate the sums in Exercises 23鈥�28.

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

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

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:

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

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

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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