the two polar cruves r=1 and r=theta^1/2. Find the area between 0 and their first intersection point

Question

when graphed the root theta is a loop like a lolipop and the 1 is just a circle. What is the area between. I would set the integral bounds from 0 to the first intersection point, but that intersection point when i try to calculate it is simply pi/4 right?

Answer 1

The intersection is at SQRT(ϴ)=1, which gives ϴ=1.

Area between the curves from ϴ=0 to ϴ=1 is an integral from 0 to 1 of 1/2((outer R)^2 -(inner r)^2)dϴ

integral from 0 to 1 of 1/2(1 -ϴ)dϴ = 1/2 ϴ-1/4ϴ^2 from 0 to 1 = 1/2 - 1/4 -0=1/4

Answer 2

Jacob:

The curves intersect at θ=1, r=1

You can first integrate using element of area in polar coordinates (rdθdr)

over θ from 0 to r² [θ(r)=r² (from r=sqrt(θ))] then integrate the result (which is r²) over r from 0 to 1 . So your integral would be of r³ from 0 to 1 which is 1/4

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