Coordinate geometry › Polar and parametric curves

Areas in polar coordinates

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1. Area bounded by polar sec and csc curves ●○○○○ 10 ptsFind the area of the region bounded by the graph of $r = \sec \theta$, the graph of $r = \csc \theta$, the $x$-axis, and the $y$-axis.
2. 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.
3. Area of a triangle in polar coordinates ●○○○○ 10 ptsIn the polar coordinate system, points $A\left(2, \frac{\pi}{6}\right)$ and $B\left(4, \frac{\pi}{3}\right)$ are given, and $O$ is the pole. Find the area $S$ of $\triangle AOB$.
4. Area enclosed by a lemniscate ●●○○○ 20 ptsUse a double integral in polar coordinates to find the area of the region enclosed by the lemniscate $r^2=4\cos(2\theta)$. Include both lobes.
5. Area enclosed by a parametric curve ●●○○○ 20 ptsFind the area of the region enclosed by the parametric curve $x = 6\cos^2 t$, $y = 6\cos t\sin t$. Round to two decimal places.
6. Area of one rose petal ●●○○○ 20 ptsFind the area enclosed by a single loop of the polar curve $r=4\cos(3\theta)$. Round to two decimal places.
7. 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.
8. Area under a rotating rectangle's path ●●●○○ 50 ptsA $2 \times 3$ rectangle has vertices $(0,0)$, $(2,0)$, $(0,3)$, $(2,3)$. Rotate it $90^{\circ}$ clockwise about $(2,0)$, then $90^{\circ}$ clockwise about $(5,0)$, then $90^{\circ}$ clockwise about $(7,0)$, and finally $90^{\circ}$ clockwise about $(10,0)$, so that the side originally on the $x$-axis is back on the $x$-axis. A point of the rectangle initially at $(1,1)$ traces a curve during these rotations. Find the area of the region below this curve and above the $x$-axis. Round to two decimal places.
9. Area swept by a rotating cardioid ●●●○○ 50 ptsThe polar curve $r = 1 + \cos \theta$ is rotated once around the point with polar coordinates $(2,0)$. What is the area of the region it sweeps over? Round to two decimal places.

More in Polar and parametric curves

Converting coordinatesParametric and polar conicsTangents to polar curves