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.
y = (1+x²)^−1,y = 0,x = 0, and x = 2; about the y-axis
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
a. What is the radius of a cylindrical shell at a point x in [0, 4]?
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
b. What is the height of a cylindrical shell at a point x in [0, 4]?
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
c. Write an integral for the volume of the solid using the shell method.
Assume f and g are continuous, with f(x) ≥ g(x) ≥ 0 on [a, b]. The region bounded by the graphs of f and g and the lines x=a and x=b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.
Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.
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
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
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 = √x,y=0, and x=4; about the x-axis
Consider the region R in the first quadrant bounded by y=x^1/n and y=x^n, where n>1 is a positive number.
a. Find the volume V(n) of the solid generated when R is revolved about the x-axis. Express your answer in terms of n.
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.
b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.
d. Write an integral for the volume of the solid.
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.
b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.
c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.
d. Write an integral for the volume of the solid.