Determine the area common to r= 3+3sintheta and r=3

Question

Please help.

The instruction says the answer must be in Exact form.

Determine the area common to r= 3 + 3sintheta and r= 3

when I graphed this it is a cardioid and a circle, I am having hard time starting to solve for this because when i try to fin the intersection using r1+r2 it givesss weird results

our topic right now is about polar curves and polar regions.

thank you so much

Answer 1

If you plot the polar graph of the two functions and observe what is "in common" between the two curves. The top part is a semicircle, A= πr²/2 and the bottom part is the double humps. So the hard part is to find the area of the double hump portion below the x-axis. This portion of 3+3sinθ goes from θ=π to 2π.

So the area of the bottom part is (1/2)∫r²(θ)dθ integrated from π to 2π.

Then add that value to the semicircle area.

Answer 2

The best way to look at this problem is by graphing it on a spiral graph with these given polar coordinates r and theta. For the first graph is not involved in the sin part of the function because b is the exponent in front of the sin(theta) and b= 0 in this r = 3 spiral graph!!

r = radius and all you need is angle which in this case is in radians (portion of pi)

Also it is good to just find the intersection points of the 2 functions

set equations equal to each other to find the r, then plug it into one of the original equations to find theta and now you have the 2 intersection points on these 2D polar functions!!

They intersect at points (3, 0) and (-3, pi) where 0 in the first point is in radians like the 2nd y coordinate pi is in radians (which is 180 degrees converted to radians)

Think Trig, not degrees !! I mean Unit Circle as a corresponding situation with sin and radians in terms of pi!!

And the area between will have a "range" of (0,3) (for y values if it is an x,y graph) and in the vertical (pi/2) direction a "domain" of (-3,3) (for x values if it is a x,y graph)

And of course it will still keep that cardoid shape on the bottom "humps" !!