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

y = ln (x−√x²−1), for 1 ≤ x ≤ √2(Hint: Integrate with respect to y.)​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = √x,y=0, and x=4; about the x-axis​​

Determine the area of the shaded region in the following figures.

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.

v(t)=2 sin t, for 0≤t≤π

Emptying a partially filled swimming pool If the water in the swimming pool in Exercise 35 is 2 m deep, then how much work is required to pump all the water to a level 3 m above the bottom of the pool?

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x,y = 2x+2,x = 2, and x=6; about the y-axis

Orientation and force A plate shaped like an equilateral triangle 1 m on a side is placed on a vertical wall 1 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then compute the force on each plate.

Determine the area of the shaded region in the following figures.

64–68. Shell method Use the shell method to find the volume of the following solids.

A hole of radius r≤R is drilled symmetrically along the axis of a bullet. The bullet is formed by revolving the parabola y = 6(1−x²/R²) about the y-axis, where 0≤x≤R.​​

Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.78 if the tank is full of gasoline. The density of gasoline is ρ = 737 kg/m³.

Lengths of symmetric curves Suppose a curve is described by y=f(x) on the interval [−b, b], where f′ is continuous on [−b, b]. Show that if f is odd or f is even, then the length of the curve y=f(x) from x=−b to x=b is twice the length of the curve from x=0 to x=b. Use a geometric argument and prove it using integration.

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x² and y = 2−x²; about the x-axis

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = e^−t; v(0) = 60; s(0) = 40

Find the area of the region described in the following exercises.

The region bounded by y=x^2−2x+1 and y=5x−9

A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0≤h≤8 0 ≤ ℎ ≤ 8 . Find the volume of water in the bowl as a function of h. (Check the special cases h=0 and h=8.)

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=4−x^2,x=2, and y=4; about the y-axis

Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

Determine the area of the shaded region in the following figures.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=x,y=2x, and y=6 ; about the y-axis​​

64–68. Shell method Use the shell method to find the volume of the following solids.

The solid formed when a hole of radius 3 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9

Use the general slicing method to find the volume of the following solids.

The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares​​

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=x^2,y=2−x, and y=0; about the y-axis

Functio​​n defined as an integral Write the integral that gives the length of the curve y = f(x) = ∫₀^x sin t dt on the interval [0,π]

Filling a spherical tank A spherical water tank with an inner radius of 8 m has its lowest point 2 m above the ground. It is filled by a pipe that feeds the tank at its lowest point (see figure). Neglecting the volume of the inflow pipe, how much work is required to fill the tank if it is initially empty?

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = cos2t; v(0) = 5; s(0) = 7

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = −32; v(0)=50; s(0)=0

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 

x = x³ ,y = 1, and x = 0; about the x-axis

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis

A solid has a circular base; cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?​​

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x−x⁴,y=0; about the x-axis.