Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.4 Area and Arc Length in Polar - Preliminary Questions - Page 623: 3


The integral represents the area of the triangle $ACD$.

Work Step by Step

According to the figure, $\theta$ changes from $\pi/6$ (which is the angle between AC and the x-axis) to $\pi/2$ and $r=\csc \theta $ represents DC. Hence, the integral $$ \frac{1}{2} \int_{\pi / 6}^{\pi / 2} \csc ^{2} \theta d \theta $$ represents the area of the triangle $ACD$.
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