areacalculus
I'm trying to teach myself some elements of calculus in preparation for my class next semester, but I'm not sure how to work this problem. I've always had trouble dealing with areas inside of shapes. Can anyone show me the steps to work through this sample problem I found?
Find the area of the shaded region
r = 1 + sin(Θ)
Differential equations is a rather immense subject. In spite of the risk of overwhelming you with the amount of information, I recommend looking in the Princeton Companion to Mathematics, from which the relevant sections are (page numbers are within parts)
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The area is equal to difference between the area of two lenses.
It is easy to find the area of lenses like the one I did in this question before: How to find the shaded area
Best Answer
Here's a start.
Your integral goes from $r=0 \to 1+\sin\theta$ and $\theta = \pi/2 \to \pi.$
$$A = \int_{\pi/2}^{\pi} \int_0^{1+\sin\theta} r\;dr\;d\theta \\ = \frac12 \int_{\pi/2}^{\pi}(1+2\sin\theta + \sin^2 \theta) d\theta.$$
Can you take it from there?