Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln 5x, y = 2, y = 3, x = 0; about the y-axis

2.Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y2 = 2x, x = 2y; about the y-axis

3.Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = x, y = 0, x = 2, x = 7; about x = 1

1. V = 15.95 (to 2 decimal places)

2. V = 107.23 (to 2 decimal places)

3. V = 560.25 (to 2 decimal places)

Step-by-step explanation:

1. y = ln 5x, y = 2, y = 3, x = 0; about the y-axis

Find volume using the disk method.

First find inverse of y=ln(5x)

5x = exp(y)

x(y)=exp(y)/5

Width of each strip = dy

length of each strip = x(y)

volume of each disk by rotation of strip about y=axis

dV = 2*pi*x(y)dy

total volume  

V = integral (dV) for y=2 to 3

= integral (2*pi*e^y/5) for y=2 to 3

= 2*pi*(e^y/5) for y=2 to 3

= 2pi(e^3-e^2)/5

= 15.95 (to 2 decimal places)

2. y2 = 2x, x = 2y; about the y-axis

Find point of intersection between  

solve y^2/2 = 2y  =>  y=4, x=2y=8, therefore

intersection is at (8,4), which is the upper integration limit

Using the disk method again

Volume of each disk

dV(y) = pi((2y)^2-(y^2/2)^2)dy

Total volume of solid  

V = integral(pi((2y)^2-(y^2/2)^2)dy) for y=0 to 4

= pi (4y^3/3 - y^5/20)  for y = 0,4

= pi (256/3 - 1024/20)

= 512pi/15

= 107.23 (to 2 decimal places)

3. y = x, y = 0, x = 2, x = 7; about x = 1

Use the shell method.

volume of each shell formed by roatation of a vertical strip about the axis of rotation (x=1)

dV = 2*pi*(x-1)*(y*dx)

Total volume of rotation

V = integral(2*pi*(x-1)*y dx for x=2 to 7

= 535pi/3

= 560.25 (to 2 decimal places)

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation:

square root

step-by-step explanation: