Page 377 Questions 13 and 15

A region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes for questions 13 and 15.

The Shell Method: Let a solid be formed by revolving a region R, bounded by x = a and x = b, around a vertical axis. Let r(x)represent the distance from the axis of rotation to x (i.e., the radius of a sample shell) and let h(x) represent the height of the solid at x (i.e., the height of the shell). The volume of the solid is

\(V = 2 \pi \int _{a} ^{b} r(x)h(x)dx\)

r(x) is the radius of the shape. h(x) is the height of the shape. The best way to imagine the equations is to graph the lines or points on a graph and imagine their 3D shape when rotated around an axis.

Question 13

Region bounded by:\(y = \sqrt x\), \(y = 0\), and \(x = 1\). Rotate about:

  1. the y-axis
  2. x = 1
  3. the x-axis
  4. y = 1
function1 <- function(x) {x * sqrt(x)}
a <- 2 * pi * integrate(function1, 0, 1)$value
function2 <- function(x) {(1 - x) * sqrt(x)}
b <- 2 * pi * integrate(function2, 0, 1)$value
function3 <- function(y) {y * (1 - (y * y))}
c <- 2 * pi * integrate(function3, 0, 1)$value
function4 <- function(y) {(2 - y) * (y * y)}
d <- 2 * pi * integrate(function4, 0, 1)$value

a
## [1] 2.513274
b
## [1] 1.675517
c
## [1] 1.570796
d
## [1] 2.617994

Question 15

The triangle with vertices (1,1), (1,2), and (2,1). Rotate about:

  1. the y-axis
  2. x = 1
  3. the x-axis
  4. y = 2
function1 <- function(x) {(2 - x) * (x)}
a <- 2 * pi * integrate(function1, 1, 2)$value
function2 <- function(x) {(2 - x) * (x - 1)}
b <- 2 * pi * integrate(function2, 1, 2)$value
function3 <- function(y) {(y) * (2 - y)}
c <- 2 * pi * integrate(function3, 1, 2)$value
function4 <- function(y) {(2 - y) * (2 - y)}
d <- 2 * pi * integrate(function4, 1, 2)$value

a
## [1] 4.18879
b
## [1] 1.047198
c
## [1] 4.18879
d
## [1] 2.094395

Answers:

Question 13

  1. 2.513274
  2. 1.675517
  3. 1.570796
  4. 2.617994

Question 15:

  1. 4.18879
  2. 1.047198
  3. 4.18879
  4. 2.094395