(a) Explanation Given: The function, f ( x , y ) = 1 ( x 2 + y 2 ) n 2 . The region D lies between the circles centered at origin and the radii r and R , respectively. Formula used: If f is a polar rectangle R given by 0 ≤ a ≤ r ≤ b , α ≤ θ ≤ β , where 0 ≤ β − α ≤ 2 π , then, ∬ R f ( x , y ) d A = ∫ α β ∫ a b f ( r cos θ , r sin θ ) r d r d θ (1) If g ( x ) is the function of x and h ( y ) is the function of y then, ∫ a b ∫ c d g ( x ) h ( y ) d y d x = ∫ a b g ( x ) d x ∫ c d h ( y ) d y (2) Calculation: The radius of the smaller circle is denote the letter r. so, use the variable t for the polar coordinates instead of r . From the given region D , it is observed that the value of t varies from r to R and the value of θ varies from 0 to 2 π . Substitute x = t cos θ and y = t sin θ (b) To determine To find: For what values of the integer n, the integral in part (a) have a limit as r → 0 + . (c) To determine To evaluate: The given integral. (d) To determine To find: For what values of the integer n, the integral in part (c) have a limit as r → 0 + .

Calculus, Early Transcendentals, International Metric Edition

8th Edition

ISBN: 9781305272378

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 15, Problem 60RE

(a)

To determine

To evaluate: The given integral.

(b)

To determine

To find: For what values of the integer n, the integral in part (a) have a limit as r→0+.

(c)

To determine

To evaluate: The given integral.

(d)

To determine

To find: For what values of the integer n, the integral in part (c) have a limit as r→0+.

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Chapter 15, Problem 59RE

Chapter 15, Problem 1P

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Chapter 15 Solutions

Calculus, Early Transcendentals, International Metric Edition

Ch. 15.1 - (a) Estimate the volume of the solid that lies...Ch. 15.1 - If R = [0, 4] [1, 2], use a Riemann sum with m =...Ch. 15.1 - (a) Use a Riemann sum with m = n = 2 to estimate...Ch. 15.1 - (a) Estimate the volume of the solid that lies...Ch. 15.1 - Let V be the volume of the solid that lies under...Ch. 15.1 - A 20-ft-by-30-ft swimming pool is filled with...Ch. 15.1 - A contour map is shown for a function f on the...Ch. 15.1 - The contour map shows the temperature, in degrees...Ch. 15.1 - Evaluate the double integral by first identifying...Ch. 15.1 - Evaluate the double integral by first identifying...

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The given integral .

Solution Summary: The author evaluates the value of the given integral, which lies between the circles centered at origin and the radii r and R.

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To evaluate: The given integral.

To find: For what values of the integer n, the integral in part (a) have a limit as r→0+.

To evaluate: The given integral.

To find: For what values of the integer n, the integral in part (c) have a limit as r→0+.

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Chapter 15 Solutions