What the heck does this question mean? They're just two 4-leaf clovers shifted 45\(^o\) (aka. \(\frac{\pi}{4}\)) relative to each other...
Find the area of the region that lies inside both curves.
r = sin 2θ, r = cos 2θ
Polar coordinates, and looking at the graphs I'm not sure what is meant here...

- anonymous

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- anonymous

|dw:1343424196418:dw|

- anonymous

Hideous graph I know but it's the best I could do relatively quickly free-handed with a mouse.

- anonymous

very Compatible topic
What the heck does this question mean?
....:)

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## More answers

- anonymous

Eh?
I'm asking, what are they looking for me to find the area of. Typically this "in-between the curves" area type problems involved two circles or a circle and a carotid, or something else. I'm not really sure what "sections" of area they are looking for with the question worded like so for these very similar functions.

- anonymous

|dw:1343424564173:dw|

- anonymous

That's what I was wondering, find the area of all 8 overlaps? O_o I seriously DON'T have any more information than what I posted at the start of this question. It's just in a section of challenge problems for polar stuff.

- anonymous

"Find the area of the region that lies inside both curves. r = sin 2θ, r = cos 2θ" That's it. Literally.

- anonymous

me too...i have any idea...almost forgot these polar stuff.

- anonymous

you'll probably have to use symmetry and just find the area of just one of those "intersections" then multiply by 8|dw:1343425073279:dw|

- anonymous

I'm trying to figure out when they intersect I guess, right?
sin 2θ = cos 2θ
So every \(\frac{\pi}{8}\) radians?

- anonymous

yes... i think we should start with that...

- anonymous

Integrate for one from 0 to \(\frac{\pi}{8}\) and then multiply by 8?

- anonymous

I mean I'm out of ideas, there's no "outer" and "inner" function here, they're radially equivalent for symmetry.

- anonymous

no... don't multiply by eight now.. just find the area of that intersection first...
yeah... and me too, i'm very rusty with polar equations....:(

- anonymous

|dw:1343425260518:dw|

- anonymous

\[8 \int\limits_0^{\pi/8} \frac{1}{2} (\sin^2(2 x)-\cos^2(2 x)) \ \ \ dx = -\frac{1}{8}\]
???

- anonymous

ack or -1

- anonymous

cos(2theta) is the top graph
sin(2theta) is the bottom graph

- anonymous

top - bottom?

- anonymous

II for \( \sin 2\theta \) from 0 to \(\frac{\pi}{8} \)
and
I for \( \cos 2\theta \) from \(\frac{\pi}{8} \) to ....?

- anonymous

?= \( \frac{\pi}{4} \) ?

- anonymous

i think u should evaluate area of I and II separatly...

- anonymous

Well I mean that as your limits for 0 to \(\frac{\pi}{16}\)

- anonymous

|dw:1343425734006:dw|

- anonymous

according to what dpalnc said cos(2theta) is the top graph (I) bounds from pi/8 to pi/4 and sin(2theta) is the bottom graph (II) bounds from 0 to pi/8

- anonymous

so from... \(\frac{\pi}{8}\) to... \(\frac{\pi}{8}\) + \(\frac{\pi}{8}\) Basically?

- anonymous

from pi/8 to ( where \( \cos 2 \theta=0 \) )

- anonymous

|dw:1343426180407:dw|

- anonymous

so i think u should set up ur integrals like this:
A=Total Area=8\(( A_I + A_{II} )\)

- anonymous

\[A=\frac{8}{2} \left( \int\limits_{0}^{\frac{\pi}{8}} \sin^2 2 \theta \ d \theta+\int\limits_{\frac{\pi}{8}}^{\frac{\pi}{4}} \cos^2 2 \theta \ d \theta \right)\]

- anonymous

|dw:1343505800216:dw|

- anonymous

So the area of the entire shaded region is obtained by :
\[\huge A=16\int_\frac{\pi}{8}^\frac{\pi}{4} \frac{1}{2}cos(2\theta)d\theta \]
=\(\huge \frac{\pi - 2}{2} \)

- anonymous

Yay @dpaInc ! That was correct, and that method worked for the second part too

- anonymous

glad to hear it....:)
i was really rusty with polar so i reviewed this a bit....
what i said earlier about "top" minus "bottom" function does not apply in polar coordinates......
it is "outer function" minus "inner function" when dealing with polar graphs....

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