How do I integrate rcos2theta dr? and then I have to plug in the limits 1 and 2sin2theta for theta and plug what I get from that into a second integral. Help!

To integrate the expression rcos^2θ dr, you can follow these steps:

Step 1: Simplify the expression
Since cos^2θ can be written as (1/2)(1 + cos2θ), the expression rcos^2θ can be simplified as (1/2)r(1 + cos2θ).

Step 2: Separate variables
Decompose the expression into two separate terms, one with r and the other with θ, so we have: (1/2)r(1 + cos2θ) dr.

Step 3: Integrate with respect to r
To integrate with respect to r, treat r as a constant and integrate the expression (1 + cos2θ) dr. The integration of (1 + cos2θ) with respect to r is simply (1 + cos2θ)r.

Step 4: Integrate the result with respect to θ
Now that we have integrated with respect to r, we are left with an expression (1 + cos2θ)r. We need to integrate this expression with respect to θ, but before doing that, it is necessary to substitute the limits of integration for θ.

Step 5: Substitute the limits for θ
The limits for θ are given as 1 and 2sin^2θ. Substitute these values into the expression (1 + cos2θ)r:
∫[(1 + cos2θ)r] dθ from 1 to 2sin^2θ.

Step 6: Evaluate the integral
Now that we have substituted the limits, we can evaluate the integral by plugging in the upper limit (2sin^2θ) minus the value of the lower limit (1).

Finally, evaluate the result of the integral at 2sin^2θ and subtract the result for 1.