Polar coordinates finding area between two curves

th3plan

Homework Equations

r=sinx
r= cosx

Ok , i need help how to properly select the integral to evaluate the area they make. Can someone please show me how , i know how to evaluate it just having hard times with integrals

The Attempt at a Solution

EngageEngage
it will be easiest if you first draw out the curves. This will help you figure out what your integral should be.

th3plan
i graphed it on calculator, and i did set them equal to each other to get tanx=1 but from here on idk what to do

ice109
first of all over what interval?

EngageEngage
if you're given those in parametric form in polar, you are going to get two circles in the x,y (or r,theta) plane, i believe. But as ice109 said, its also important that you know how x varies for this one.

th3plan
the interval is 0,2pi

rocomath
Multiply both sides by r, then change to rectangular form. Is this Calculus 2 or 3? B/c I did this problem yesterday.

th3plan
Calc 2

th3plan
how would i change it to rectangular form, using x=rcos(theta) and y=rsin(theta)

?

No, it isn't. Since sine and cosine are negative for half that interval using 0 to 2$\pi$ gives you each circle twice. And, in fact, the area you want only requires $\theta$ going from 0 to $\pi/2$.
However, you are correct that the circles intersect when tan$\theta$= 1- that is, at $\theta= \pi/4$ as well as at 0. For $0\le \theta\le \pi/4$, a radius goes from 0 to cos($\theta$) while from $\pi/4\le \theta\le \pi/2$ it goes from 0 to sin($\theta$). From symmetry, you should be able to integrate cos($\theta$) from 0 to $\pi/4$ and double.