Polar coordinates finding area between two curves

[th3plan]

Homework Statement

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)

?

 

[HallsofIvy]

the interval is 0,2pi

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.