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Problem 6.3.48
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.
R is bounded by y=x^2 and y=√8x.
Problem 6.4.16
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 = 4 / y + y³,x = 1/√3, and y=1; about the x-axis
Problem 6.5.9
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = −8x−3 on [−2, 6] (Use calculus.)
Problem 6.2.57
Find the area of the region described in the following exercises.
The region bounded by y=x^2−2x+1 and y=5x−9
Problem 6.3.36
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| and y=2−x^2; about the x-axis
Problem 6.2.37
Find the area of the region described in the following exercises.
The region bounded by y=4x+4, y=6x+6, and x=4
Problem 6.2.15
Determine the area of the shaded region in the following figures.
Problem 6.3.26
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=x+2,x=0, and x=4 ; about the x-axis
Problem 6.4.67
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
Problem 6.3.34
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=√sin x,y=1, and x=0; about the x-axis
Problem 6.4.19
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 = 1−x²,x = 0, and y = 0, in the first quadrant; about the y-axis
Problem 6.7.41
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?
Problem 6.3.46
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.
R is bounded by y=4−2x, the x-axis, and the y-axis.
Problem 6.1.35
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
Problem 6.6.3
Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.
Problem 6.5.2
Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.
Problem 6.6.13
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
Problem 6.7.38
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?
Problem 6.4.30
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.
{Use of Tech} y = In x/x²,y = 0,x = 3, about the y-axis
Problem 6.3.40
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
Problem 6.2.25
Determine the area of the shaded region in the following figures.
Problem 6.2.29
Determine the area of the shaded region in the following figures.
Problem 6.2.19
Determine the area of the shaded region in the following figures.
Problem 6.2.23
Determine the area of the shaded region in the following figures.
Problem 6.2.27
Determine the area of the shaded region in the following figures.
Problem 6.1.72
Without evaluating integrals, prove that ∫₀² d/dx(12 sin πx²) dx=∫₀² d/dx (x¹⁰(2−x)³) dx.
Problem 6.7.7
What is the pressure on a horizontal surface with an area of 2 m² that is 4 m underwater?
Problem 6.5.17
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
Problem 6.2.49
Find the area of the region described in the following exercises.
The region in the first quadrant bounded by y=x^2/3 and y=4
Problem 6.5.14
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = x^3/2 / 3 − x^1/2 on [4, 16]
