Answer:
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)
