area enclosed by TWO curves

[iDoof]

i've got two parametric curves:

r = sin t
r = cos t

i want to find the area that the two curves share. i know what limits to use on the integration but what equation do I integrate?


thanks!!!

 

[stapel]

What are the limits of integration? Are you supposed to find the area of one loop or something? Didn't your book give you a formula for polar-coordinate integrations?

Eliz.

 

[iDoof]

well, i graphed it and it shows two oval-like curves that overlap. so it's not one of those "area under one loop" things. I have a formula for the area it's the integral from a to b of (1/2)r^2


but there are 2 r's so i don't know what to do!

 

[stapel]

but there are 2 r's so i don't know what to do!

Without being able to see the specifics of the exercise as presented in your text, it's hard to say what they're wanting, but I would think that the two "r" functions should work just like when you had two "y" functions and had to find the area in between.

Eliz.

 

[soroban]

Hello, iDoof!

i've got two parametric curves: \(\displaystyle \;r\:=\:\sin(t),\;\;r\:=\:\cos(t)\)

i want to find the area that the two curves share.
i know what limits to use on the integration but what equation do I integrate?

Are you sure you know the limits?

                |
              * * *
          *     |     *
        *       |       *
      *         |         *
                |             / θ = π/4
     *          |          */
     *          +       *:/*  *
     *          |    *::/::*     *
                |  *::/:::         *
      *         |*::/:::::*          *
        *       |:/:::::*
          *     *:::::*               *
    ----------*-*-*--------+----------*----
                *                     *
                |
                |*                   *
                |  *               *
                |    *           *
                |        * * *

For \(\displaystyle \theta = 0\) to \(\displaystyle \frac{\pi}{4}\), use \(\displaystyle r\,=\,\sin\theta\)

For \(\displaystyle \theta=\frac{\pi}{4}\) to \(\displaystyle \frac{\pi}{2}\). use \(\displaystyle r\,=\,\cos\theta\)

 

[iDoof]

ah i see now. thanks for the visual explanation too!

 

[stapel]

thanks for the visual explanation too!

soroban is a master! :shock:

Eliz.