Evaluate the iterated integral by converting to polar coordinates. Round your answer to four decimal places.
A) 10.5742
B) 13.5742
C) 17.5742
D) 28.5742
E) 14.5742

Question

Evaluate the iterated integral  by converting to polar coordinates. Round your answer to four decimal places.
A) 10.5742 
B) 13.5742 
C) 17.5742 
D) 28.5742 
E) 14.5742

Quizplus · Accepted Answer

Converting to polar coordinates, we have:

$$\int_{0}^{1} \int_{\sqrt{x}}^{1}\sqrt{x^2+y^2}dydx = \int_{0}^{\frac{\pi}{4}}\int_{0}^{\sec{	heta}}r\cdot r dr d	heta = \int_{0}^{\frac{\pi}{4}}\frac{1}{3}\sec^3{	heta}d	heta$$

Evaluating the integral, we get:

$$\int_{0}^{\frac{\pi}{4}}\frac{1}{3}\sec^3{	heta}d	heta = \frac{1}{3}\left[\frac{\sec{	heta}(\sec{	heta}+	an{	heta})}{2}ight]_{0}^{\frac{\pi}{4}} = \frac{1}{3}\left[\frac{\sqrt{2}+1}{2} + \frac{1}{2}ight] = \frac{1}{3}\left[\frac{\sqrt{2}+3}{2}ight] \approx 10.5742$$

Rounding this to four decimal places, we get the final answer as A) 10.5742.

Evaluate the Iterated Integral by Converting to Polar Coordinates