9. Graphical Applications of Integrals

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = y⁴/4 + 1/8y², for 1≤y≤2

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 $y=x+4$, $y=0$, $x=1$ & $x=5$.

Find the volume of the solid formed by revolving the area bounded by $f\left(x\right)=x^2$ from $x=0$ to $x=3$ and the y-axis around the y-axis.

74. Volume of a Solid

Consider the region R bounded by:

The graph of f(x) = 1/(x + 2)

The x-axis on the interval [0,3].

Find the volume of the solid formed when R is revolved about the y-axis.

Let f(x) = √(x + 1). Find the area of the surface generated when:

Region bounded by f(x) and the x-axis on [0, 1]

Revolved about the x-axis

65. Volume Find the volume of the solid generated when the region bounded by y = sin²(x) * cos^(3/2)(x) and the x-axis on the interval [0, π/2] is revolved about the x-axis.

87. Surface area Find the area of the surface generated when the curve f(x) = sin x on [0, π/2] is revolved about the x-axis.

Find the area of the surface generated when the given curve is revolved about the given axis.

y=3x+4, for 0≤x≤6; about the x-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=8√x, for 9≤x≤20; about the x-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=(3x)^1/3 , for 0≤x≤8/3; about the y-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=√1−x^2, for −1/2≤x≤1/2; about the x-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=4x−1, for 1≤x≤4; about the y-axis (Hint: Integrate with respect to y.)

Find the area of the surface generated when the given curve is revolved about the given axis.

y=1/4(e^2x+e^−2x), for −2≤x≤2; about the x-axis