Volumes of Revolutions

[paulxzt]

1) y = x^2/3, x = 1 , y= 0 about the y-axis

my solution: the integral from 0 to 1 of pi(y^3/2)^2
= .785

2) y = x, y = sqrt(x) , about x=2
my sol: integral from 0 to 1 of pi (2 - sqrt(x))^2 - pi(x)^2 dx

are my solutions correct? I think i got the bounds wrong on #1 i am not sure..
for #2, is my setup correct?

 

[skeeter]

1) y = x^2/3, x = 1 , y= 0 about the y-axis

my solution: the integral from 0 to 1 of pi(y^3/2)^2
= .785 correct


2) y = x, y = sqrt(x) , about x=2
my sol: integral from 0 to 1 of pi (2 - sqrt(x))^2 - pi(x)^2 dx

pi*integral{0 to 1} [2 - sqrt(x)]^2 - (2 - x)^2 dx

are my solutions correct? I think i got the bounds wrong on #1 i am not sure..
for #2, is my setup correct?

 

[galactus]

You can also use shells on the first one.

\(\displaystyle \L\\2{\pi}\int_{0}^{1}[x(1-x^{\frac{2}{3}})]dx=\frac{\pi}{4}\)


For the second, I get:

washers: \(\displaystyle \L\\{\pi}\int_{0}^{1}[(y^{2}-2)^{2}-(y-2)^{2}]dy\)

shells: \(\displaystyle \L\\2{\pi}\int_{0}^{1}(x-2)(x-\sqrt{x})dx\)