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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 6.1.62
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
Problem 6.4.65
64–68. Shell method Use the shell method to find the volume of the following solids.
A right circular cone of radius 3 and height 8
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 6.2.63
Find the area of the shaded regions 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 6.2.15
Determine the area of the shaded region in the following figures.
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 6.7.5
Why is integration used to find the work required to pump water out of a tank?
Problem 6.3.38
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
Problem 6.2.51
Find the area of the region described in the following exercises.
The region bounded by y=e^x, y=2e^−x+1, and x=0
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 6.6.13
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 6.3.40
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 6.2.17
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 6.7.2
Explain how to find the mass of a one-dimensional object with a variable density ρ.
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]
