Question

Sketch the region of integration, carefully with appropriate labels, and then perform the following integral: ∫(0 to pi/4)∫(0 to 4secO) r dr dO

Answer 1

$\int _0^{\frac {\pi}{4}}d \theta \int_0^{4\sec \theta}rdr$ $=\frac {1}{2}\int_0^{\frac{\pi}{4}}16\sec^2\theta d\theta$ $=[\sec \theta =\frac{1}{\cos\theta}]$ $=8\tan\theta|_0^{\frac {\pi}{4}}$ $=8$ The sketch is attached below.