102–105. Volumes 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 generated when R is revolved in the following ways.
102. About the y-axis

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:

a. Apply the disk method and integrate with respect to y.

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:

b. Apply the shell method and integrate with respect to x.

Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).

b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.

69-72. Volumes of solids Find the volume of the following solids.

70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.

102–105. Volumes 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 generated when R is revolved in the following ways.

104. About the line y = 1

Find the volume of the solid whose base is the region bounded by the function $f\left(x\right)=\sqrt{9-x^2}$ and the x-axis with square cross sections perpendicular to the x-axis.

Find the volume for a solid whose base is the region between the curve $y=\sqrt{\sin x}$ and the x-axis on the interval from $[0,\pi ]$ and whose cross sections are equilateral triangles with bases parallel to the y-axis.

Find the volume of the solid obtained by rotating the region bounded by $y=x+4$, $y=0$, $x=1$ & $x=5$ about the x-axis.