A solid S is generated by revolving the region between the x-axis and the curve y=

$$\sqrt { \sin x } ( 0 \leq x \leq \pi )$$

about the x-axis. An integral expression for the volume of S is ___.

Find the area of the surface obtained by revolving the curve x=sin t, y=cos 2t, for $-\pi / 4 \leq t \leq \pi / 4$, about the x-axis.

Let R be the region in the first quadrant enclosed by $y=x^2$, $y=2+x$, and x=0. Calculate the volume of the solid generated by revolving R about the x-axis by integrating with respect to x.

Find the area of the surface generated by revolving the curve $x=2+\cos t, y=1+\sin t$, for $0 \leq t \leq 2 \pi$ about the x-axis.

Graph the parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range.

$$x=-2(y+3)^2$$

Use the result of the previous part to find the volume of the solid generated by revolving each plane region about the y-axis. (Hint: Begin by approximating the points of intersection.)

Graph the parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range.

$$x=(y-3)^2$$

Graph the parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range.

$$x=y^2+2 y-8$$

Find the surface area of the solid generated by revolving each curve about the x-axis. $x(t)=3 t^{2}$, $y(t)=6 t$; $0 \leq t \leq 1$

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. $x y=1, \quad 1 \leq y \leq 2 ; \quad y \text {-axis }$

$\rule{1cm}{0.4pt}$ is an example of an inflammatory mediator that stimulates chemotaxis. a. Endotoxin $\hspace{2cm}$ c. A fibrin clot b. Serotonin $\hspace{2cm}$ d. Interleukin-2

Find the volume generated by revolving the regions bounded by the given curves about the $x$-axis. Use the indicated method in each case.

$y=x^2+1, y = x + 1$ (shells)

Find an equation of a curve given that it passes through the point $\left(2, \frac{2\sqrt{5}}{3}\right)$ and that the slope of the tangent line to the curve at any point $P(x,y)$ is given by

$$\frac{dy}{dx}=-\frac{4x}{9y}$$

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the y-axis. $y=\frac{x^{2}}{2}+4$, $0 \leq x \leq 2$