Having problems with a homework question:
Part A asks to sketch the solid region bounded in the first octant by elliptic cylinder 2x^2 + y^2 =1, and the plane y + z = 1.
I know we have to do this question with cylindrical coordinates; thus, after substituting x = rcos(theta) and y = rsin(theta), I get:
z = 1- rsin(theta)
r = 2/(3 + cos2(theta)); after the substitution sin^2(theta) = 1 - cos^2(theta), and cos^2(theta) = (1 + cos2(theta))/2.
I have set up limits of integration as follows:
for d(theta), from 0 to pi/2 because of the first quadrant requirement
for dr from 0 to r = 2/(3 + cos2(theta))
for dz from 0 to z = 1- rsin(theta)
Of course I have not forgotten that dV = rdrd(theta)dz
I get stuck with an ugly trig expression when I get to the integration for d(theta)
What also confuses me is that the graph for the elliptic cylinder is not continuous, that is y = (1-2x^2)^(1/2) is not joined with
y = -(1-2x^2)^(1/2)!
Another quick quesiton:
How can you integrate cos^5(theta)d(theta) without looking it up in an integration table for which the answer also includes the integration of
For the integration of sin^3(theta)d(theta) I have seen the substituion of
u = cos(theta), thus, u^2 = cos^2(theta) = 1 - ( 1- sin^2(u)) =
sin^2(theta), and du = -sin(theta)d(theta). Therefore we can integrate
(1 - u^2)du which is equivalent to sin^3(theta)d(theta).
Is there a similar way for integrating cos^5(theta)d(theta)?
Thank you very much