Question

Find the volume of the solid of revolution obtained by rotating the curve `x=3cos^3theta` , `y=3sin^3theta` about the `x` axis?

Answer 1

See the answer below:

Answer 2

See a way to do the integrals involved more quickly below.

Explanation:

I'm not going to answer the question at hand here, but I thought it would be helpful to point out that the integrals with odd powers of sine and cosine can be solved more quickly.

For example, to do `\int\ sin^{9}(theta) d\theta` quickly, it's best to write `sin^{9}(theta)=(sin^{2}(theta))^{4} * sin(theta)=(1-cos^{2}(theta))^{4} * sin(theta)`. With the substitution `u=cos(theta)`, `du=-sin(theta)d\theta`, the integral then can be solved as follows (with the help of the binomial theorem):

`int\ sin^{9}(theta)d\theta=int\ -(1-u^2)^{4}du`

`=int (-u^8+4u^6-6u^4+4u^2-1)du`

`=-1/9 u^9+4/7 u^7-6/5 u^5+ 4/3 u^3-u+C`

`=-1/9 cos^{9}(theta)+4/7 cos^{7}(theta)-6/5 cos^{5}(theta)+4/3 cos^{3}(theta)-cos(theta)+C`.

Integrals involving only even powers of sine and cosine are harder to solve quickly. The identities `cos^{2}(theta)=1/2+1/2 cos(2theta)` and `sin^{2}(theta)=1/2-1/2 cos(2theta)` are helpful, but they are still typically a lot of work.