Hi, I was wondering if someone can help me complete this multi-step problem. The parts I've done are in blue. I'll post the whole of it, first (sorry, I don't know how to post an image of the curve):
The curve above is drawn in the xy-plane and is described by the equation in polar coordinates r = θ + sin(2 θ) for 0 ≤ θ ≤ π, where r is measured in meters and θ is measured in radians. The derivative of r with respect to θ is given by dr/dθ = 1 + 2cos(2θ).
a) find the area bounded by the curve and the x-axis.
So, for this, the area formula would be 1/2 ∫ r^2 dθ, no? So it would end up being:
1/2 ∫( θ + sin(2 θ) )^2, from 0 to pi, which yields?
b) find the angle θ that corresponds to the point on the curve with x-coordinate -2.
? I thik it means that since x = r cos θ = f(θ) cos θ , you get this:
-2 = (θ + sin(2 θ)) cos θ
Is this right? Am I just supposed to solve forθ?
c) For π/3 < θ < 2π/3, dr/dθ is negative. What does this fact say about r? What does this fact say about the curve?
I am guessing that since dr/dθ is negative, 0>dr/dθ, which makes r decreasing. And because it is decreasing, the graph gets closer to the origin. Is this right?
d) Find the value of θ in the interval 0 ≤ θ ≤ π/2 that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.
? I still don't know how to do this.
I'd really appreciate any help you can offer. Thank you again!