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Problem 6.2.25
Determine the area of the shaded region in the following figures.
Problem 6.5.13
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = (x²+2)^3/2 / 3 on [0, 1]
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 6.3.31
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 xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.
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 6.7.5
Why is integration used to find the work required to pump water out of a tank?
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 6.7.9
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 top half of the tank
Problem 6.6.17
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
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 6.2.39
Find the area of the region described in the following exercises.
The region bounded by y=e^x, y=e^−2x, and x=ln 4
Problem 6.4.34
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} y² = ln x,y² = ln x³, and y=2; about the x-axis
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 6.6.7
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
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 6.1.63
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) = t(25−t²)^1/2, for 0≤t≤5
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 6.6.35
Find the area of the surface generated when the given curve is revolved about the given axis.
x=4y^3/2−y^1/2 / 12, for 1≤y≤4; about the y-axis
Problem 6.4.58
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 = x³,y=0, and x=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 6.7.11
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 6.7.56
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.
Problem 6.7.66
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.
Problem 6.7.51
Force on a triangular plate A plate shaped like an isosceles triangle with a height of 1 m is placed on a vertical wall 1 m below the surface of a pool filled with water (see figure). Compute the force on the plate.
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 6.3.55
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 6.6.3
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 6.5.14
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]
