Question

Use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. y = √x, y = 2, x = 0.
a. the x-axis
b. the line y = 2
c. the y-axis
d. the line x = -1

Answer 1

Answer:

a) 8π

b) 8/3 π

c) 32/5 π

d) 176/15 π

Step-by-step explanation:

Given lines :  y = √x, y = 2, x = 0.

a) The x-axis

using the shell method

y = √x = , x = y^2

h = y^2 , p = y

vol = ( 2π ) $\int\limits^2_0 {ph} \, dy$

     = $( 2\pi ) \int\limits^2_0 {y.y^2} \, dy$  

∴ Vol = 8π

b) The line y = 2  ( using the shell method )

p = 2 - y

h = y^2

vol = ( 2π ) $\int\limits^2_0 {ph} \, dy$

     = $( 2\pi ) \int\limits^2_0 {(2-y).y^2} \, dy$

     = ( 2π ) * [ 2/3 * y^3  - y^4 / 4 ] ²₀

∴ Vol  = 8/3 π

c) The y-axis  ( using shell method )

h = 2-y  = h = 2 - √x

p = x

vol = $(2\pi ) \int\limits^4_0 {ph} \, dx$

     = $(2\pi ) \int\limits^4_0 {x(2-\sqrt{x} ) } \, dx$

     = ( 2π ) [x^2 - 2/5*x^5/2 ]⁴₀

vol = ( 2π ) ( 16/5 ) = 32/5 π

d) The line x = -1    (using shell method )

p = 1 + x

h = 2√x

vol = $(2\pi ) \int\limits^4_0 {ph} \, dx$

Hence   vol = 176/15 π

attached below is the graphical representation of P and h