매개변수 곡선의 미적분

Find the area of the region bounded by $x=2\,{\sin }^{2}\theta ,y=2\,{\sin }^{2}\theta \,\tan \,\theta ,$ for $0\le \theta \le \frac{\pi }{2}.$

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

$x=2\,\cot \,\theta ,y=2\,{\sin }^{2}\theta ,0\le \theta \le \pi$

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

[T] $x=a\,\sin (2t),y=b\,\sin (t),0\le t<2\pi$ (the “hourglass”)

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

[T] $x=2a\,\cos \,t-a\,\sin (2t),y=b\,\sin \,t,0\le t<2\pi$ (the “teardrop”)

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

$\begin{array}{ccc}x={t}^{3}, & y={t}^{2}, & 0\le t\le 1\end{array}$

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

$\begin{array}{ccc}x=a\,{\cos }^{3}\theta , & y=a\,{\sin }^{3}\theta , & 0\le \theta \le \end{array}\frac{\pi }{2}$

[T] Use a CAS to find the area of the surface generated by rotating $x=t+{t}^{3},y=t-\frac{1}{{t}^{2}},1\le t\le 2$ about the x-axis. (Answer to three decimal places.)

Find the area of the surface generated by revolving $x={t}^{2},y=2t,0\le t\le 4$ about the x-axis.

For the following exercises, find the area of the region.

$x={t}^{2},$ $y=\ln (t),$ $0\le t\le e$

For the following exercises, find the area of the region.

$r=1-\sin \,\theta$ in the first quadrant

In Exercises 41–44, a solid of revolution is described. Find or approximate its surface area as specified.

Find the surface area of the sphere formed by rotating the circle $x=2\cos t$, $y=2\sin t$ about:

(a) the $x$-axis and (b) the $y$-axis.

In Exercises 41–44, a solid of revolution is described. Find or approximate its surface area as specified.

Find the surface area of the torus (or “donut”) formed by rotating the circle $x=\cos t+2$, $y=\sin t$ about the $y$-axis.

In Exercises 41–44, a solid of revolution is described. Find or approximate its surface area as specified.

Approximate the surface area of the solid formed by rotating the “upper right half” of the bow tie curve $x=\cos t$, $y=\sin(2t)$ on $[0,\pi/2]$ about the $x$-axis, using Simpson's Rule and $n=4$.

In Exercises 41–44, a solid of revolution is described. Find or approximate its surface area as specified.

Approximate the surface area of the solid formed by rotating the one petal of the rose curve $x=\cos t\cos(2t)$, $y=\sin t\cos(2t)$ on $[0,\pi/4]$ about the $x$-axis, using Simpson's Rule and $n=4$.