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.)

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).

Sketch $(1, −2, 3)$ and $\left(-\frac{1}{3},\frac{2}{3},-1\right)$. Why do these vectors point in opposite directions?

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 $x=2$

Identify the condition in the following diagnoses.
Vestibular neuronitis, left ear
main term: __________

Apply the product rule to find the tangent line, in slope–intercept form, of y = f(x) at the specified point. $f(x)=\left(3 x^{2}-2\right)(x-1), \text { at } x=1$

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

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

Given f(x) = (x+3)/(x-2), find f^{-1}(x) and identify the domain and range of f^{-1} . As a check, show that f(f^{-1}(x))=f^{-1}(f(x))=x.

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.

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.

$$g(f(0))$$

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 ___.

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

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

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