9. Graphical Applications of Integrals

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

d. the line y = 2

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

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

_______

y = √4y ― y² , 1 ≤ y ≤ 2 ; y-axis

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]

58–61. Arc length Find the length of the following curves.

y = 2x+4 on [−2,2] (Use calculus.)

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines. 

x = -1

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.

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]?

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.

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

Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)

b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy

Volumes

Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the curve y = 1+√x, the curve y = 1−√x, and the line x=1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.

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 

Consider the following curves on the given intervals.  

a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis. 

y=tan x , for 0≤x≤π/4; about the x-axis