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Problem 6.3.22
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=0,y=lnx,y=2, and x=0; about the y-axis
Problem 6.3.70
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.)
Problem 6.3.20
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=1 / 4√1 − x^2,y=0,x=0, and x=12; about the x-axis
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 6.3.28
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
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 6.7.11
9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)
The work required to empty the tank through an outflow pipe at the top of the tank
Problem 6.6.21
A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters.
The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis
Problem 6.7.7
What is the pressure on a horizontal surface with an area of 2 m² that is 4 m underwater?
Problem 6.4.37
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
Problem 6.6.9
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
Problem 6.2.45
Find the area of the region described in the following exercises.
The region bounded by y=2−|x|and y=x^2
Problem 6.3.62
Use calculus to find the volume of a tetrahedron (pyramid with four triangular faces), all of whose edges have length 4.
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 6.3.64
Let f(x) = {x if 0≤x≤2
2x−2 if 2<x≤5
−2x+18 if 5<x≤6.
Find the volume of the solid formed when the region bounded by the graph of f, the x-axis, and the line x=6 is revolved about the x-axis.
Problem 6.4.22
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
Problem 6.2.11
Determine the area of the shaded region in the following figures.
Problem 6.2.59
Find the area of the region described in the following exercises.
The region bounded by x=y(y−1) and y=x/3
Problem 6.7.2
Explain how to find the mass of a one-dimensional object with a variable density ρ.
Problem 6.2.7
Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.
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 6.3.55
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.
y=2 sin x and y=0 on [0,π]; about y=−2
Problem 6.7.18
13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.
ρ(x) = {1 if 0≤x≤2 {2 if 2<x≤3
Problem 6.3.1
Suppose a cut is made through a solid object perpendicular to the x-axis at a particular point x. Explain the meaning of A(x).
Problem 6.4.38
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 = 8,y = 2x+2,x = 0, and x=2; about the y-axis
Problem 6.3.12
Use the general slicing method to find the volume of the following solids.
The solid whose base is the region bounded by the curves y=x^2 and y=2−x^2, and whose cross sections through the solid perpendicular to the x-axis are squares
Problem 6.7.13
13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.
ρ(x)=1+sin x, for 0≤x≤π
Problem 6.1.69
Suppose f and g have continuous derivatives on an interval [a, b]. Prove that if f(a)=g(a) and f(b)=g(b), then ∫a^b f′(x) dx = ∫a^b g′(x) dx.
Problem 6.4.41
39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.
x =2
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
