A solid is generated by revolving about the x-axis the region bounded by the graph of the positive continuous function y = f(x), the x-axis, the fixed line x = a, and the variable line x = b, b > a. Its volume, for all b, is $b^{2}-a b.$ Find f(x).

In this exercise, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

$$y=2x^2, y=0, x=2$$

the line $x=2$

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.

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.)

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.

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

$$y=(x+3)^2-4$$

Let $R$ be the "triangular" region in the first quadrant that is bounded above by the line $y=1$, below by the curve $y=\ln x$, and on the left by the line $x=1$. Find the volume of the solid generated by revolving $R$ about the $y$-axis.

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

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

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

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 $S=\int_{a}^{b} 2 \pi f(t) \sqrt{f^{\prime}(t)^{2}+g^{\prime}(t)^{2}} d t$. 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.

Find the area of the surface generated by revolving the curve about (a) the x-axis and (b) the y-axis. x = 2 cos θ, y = 2 sin θ, 0 ≤ θ ≤ π/2

Find the volume of the solid formed by revolving the region bounded by the graphs of the given equations about the x-axis. x = 6 cos θ, y = 6 sin θ

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)

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.