Question

Two short integral questions about setting up integrals?

Answer 1

Regarding the first problem: Look up the formula for "area in polar coordinates." Mind typing it out here (or drawing it), for reference?

What shape does the polar curve r=3 have?

what shape does the polar curve 5+2sin (3*theta) have?

It just so happens that the second curve intersects the first 3 times. How would you go about determining the angles (values of theta) at which this happens?

One way to do that would be to set the two polar equations equal to each other. Then\[r=3=5+2\sin 3\theta=r\] How would you go about solving this equation for theta? You should get 3 roots. Each root locates for you where the 2 polar curves meet / intersect.

I found it very helpful to draw a figure. By all means do this on a graphing calculator if you can.

Answer 2

is it 1/2 integral from a to b r^2 d theta?

Answer 3

the formula i mean

Answer 4

I drew them out as well, just figuring out the upper and lower limits confuse me as well as just setting everything up in general

Answer 5

If at all possible, please use equation Editor, below:\[A=\frac{ 1 }{ 2 }\int\limits_{\alpha}^{\beta}r^2*d \theta\] for the clarity it affords.

Answer 6

The main challenge here is to determine the limits of integration. They are angles, not x-values. I ask you to determine where your two curves intersect / touch one another; this happens in 3 places. See my earlier comments, above.

Once you have the limits of integration for ONE of the 3 "petals" that are visible when you've graphed both polar functions on one set of axes, you bassically find 2 separate areas. First, find the area enclosed within one petal, and then subtract the partial area of the circle.