Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve

$$y = e ^ { x }$$

, and the line x = ln 2 about the line x = ln 2.

The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve $x=2 / \sqrt{y+1}$ is revolved about the y -axis to generate a solid. Find the volume of the solid.

Use the integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve

$$y = x ^ { 2 } , - 1 \leq x \leq 1$$

about the x-axis.

The region R is bounded by the curve y=ln x and the x-axis on the interval [1, e]. Find the volume of the solid that is generated when R is revolved in the following ways. About the y-axis

In this exercise, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y=4$.

$$y=\dfrac{1}{2}x^3,\quad y=4\quad x=0$$

What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?

A factor of $x-3$ in the denominator.

Let R be the region bounded by $y=\ln x$, the x-axis, and the line $x=a$, where $a\gt1$.

Find the volume $V_1(a)$ of the solid generated when R is revolved about the x-axis (as a function of a).

Recurrence to explicit formula Find an explicit formula that defines the sequence generated by the recurrence relation

$a_{n+1}=\frac{2 a_n}{a_n+1}, a_0=2$.

Write an integral that represents the area of the surface generated by revolving the curve about the x-axis. Parametric Equation: x = 1/4t², y = t + 3 Interval: 0 ≤ t ≤ 3

Find the volume generated when the region between the curves is rotated around the given axis. $x=y^{3}, y=\frac{1}{x}, x=1, \text { and } y=2$ rotated around the x-axis.

Find the area of the surface generated when the arc of the curve

$${y=\frac{1}{3} x^{3}+(4 x)^{-1}}$$

between x = 1 and x = 3 is revolved about (a.) the x-axis (b.) the y-axis

Find the surface area generated when the given polar curve is revolved about the x-axis.

$$r=1-\cos \theta, 0 \leq \theta \leq \pi$$

In this exercise, find the volume of the solid generated by revolving the region bounded by the graphs of the equationsabout the $x$-axis.

$$y=x^2+1,\quad y=-x^2+2x+5,\quad x=0,\quad x=3$$

Use shells to find the volume generated by rotating the regions between the given curve and y = 0 around the x-axis. $x=\frac{1}{1+y^{2}}, y=1, \text { and } y=4$