Back9-34. 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
Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).
b. How is this formula changed if x0>b?
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
Determine whether the following statements are true and give an explanation or counterexample.
c. Let f(x)=12x^2. The area of the surface generated when the graph of f on [−4, 4] is revolved about the x-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the x-axis.
Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.
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
Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.
c. Write an integral for the volume of the solid using the shell method.
Let f(x) = √(x + 1). Find the area of the surface generated when:
Region bounded by f(x) and the x-axis on [0, 1]
Revolved about the x-axis
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]
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
Volumes
Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about
a. the x-axis
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.
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
Volumes
Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 about
c. the line x = 1
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