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2 problems match “circular sectors”. Clear

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1. Area of a polar sector ●○○○○ 10 ptsFind the area of the region given in polar coordinates by $\frac{\pi}{4}\leq\theta \leq\frac{\pi}{2}$ and $0\leq r\leq 2$, using a double integral. Round to two decimal places.
2. Area swept by a bent segment ●●●○○ 50 ptsPlot points $A, B, C$ at coordinates $(0,0)$, $(0,1)$ and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise $45^\circ$ about point $A$. Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise $45^\circ$ about point $A$. Find $\frac{X_{1}+X_{2}}{2}$. Round to two decimal places.