Write in the blank whether the sentence is simple or compound.

Lauren is sweeping the dugout, and Kat is cleaning the bases.

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

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

Find all vertical asymptotes. $f(x)=\frac{x^{2}+1}{x^{3}+3 x^{2}+2 x}$

Find the volume of the solid generated by revolving the region in the irst quadrant bounded by the coordinate axes and the curve

$$y = \cos x , 0 \leq x \leq \pi / 2$$

about

the line

$$x = \pi / 2.$$

Let C be the curve x=f(t), y=g(t), for $a \leq t \leq b$, where f' and g' are continuous on [a. b] and C does not intersect itself, except possibly at its endpoints. If g is nonnegative on [ a, b ], then the area of the surface obtained by revolving C about the x-axis is $S=\int_{a}^{b} 2 \pi g(t) \sqrt{f^{\prime}(t)^{2}+g^{\prime}(t)^{2}} d t$. Likewise, if f is nonnegative on [a, b], then the area of the surface obtained by revolving C about the y-axis is $S=\int_{a}^{b} 2 \pi f(t) \sqrt{f^{\prime}(t)^{2}+g^{\prime}(t)^{2}} d t$. These results can be derived. Find the area of the surface obtained by revolving one arch of the cycloid x=t-sin t, y=1-cos t, for $0 \leq t \leq 2 \pi$, about the x-axis.

In this exercise, you will evaluate the composite function

Let $f(x)=\sqrt{x}$ and $g(x)=x^2-1$. Evaluate the expression given below.

$$g(f(0))$$

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

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

In this exercise, find the volume of the solid generated by revolving the plane region bounded by the equations about the indicated line(s).

$$y=\sqrt{x}, y=2, x=0$$

the $x$-axis

A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of y= 1/8 $x^2 \sqrt{2-x}$ and the x-axis about the x-axis, where x and y are measured in meters. Find the tank's volume.

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. $y=\frac{x^{2}}{2}+4$, $0 \leq x \leq 2$

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

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

Let C be the curve x = f(t), y = g(t), for $a \leq t \leq b$, where f' and g' are continuous on[a, b] and C does not intersect itself, except possibly at its endpoints. If g is nonnegative on [a, b], then the area of the surface obtained by revolving C about the x-axis is $s=\int_{a}^{b} 2 \pi g(t) \sqrt{f^{\prime}(t)^{2}+g^{\prime}(t)^{2}} d t$. Likewise, if f is nonnegative on [a, b], then the area of the surface obtained by revolving C about the y-axis is $S=\int_{a}^{b} 2 \pi f(t) \sqrt{f^{\prime}(t)^{2}+g^{\prime}(t)^{2}} d t$. Find the area of the surface obtained by revolving the curve $x=\cos ^{3} t, y=\sin ^{3} t$, for $0 \leq t \leq \pi / 2$, about the x-axis.

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

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

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)