A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
a. Use the shell method to write an integral for the volume of the torus.
A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
a. Use the shell method to write an integral for the volume of the torus.
A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
b. Use the washer method to write an integral for the volume of the torus.
Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].
c. Find the volume of the solid generated when R is revolved about the x-axis.
Surface area of a cone Find the surface area of a cone (excluding the base) with radius 4 and height 8 using integration and a surface area integral.
21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = 1/x, for 1 ≤ x ≤ 10
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
Arc length may be negative if f(x) < 0 on part of the interval in question.
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=2x,y=0 , and x=3; about the x-axis (Verify that your answer agrees with the volume formula for a cone.)
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
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
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 and y=4√x; about the x-axis
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
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
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.
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
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. If a region is revolved about the y-axis, then the shell method must be used.