<p>Let R be the region inside the graph of the polar curve r =2 and outside the graph of the polar curve r=2(1-sin(theta))</p>
<p>My answer is 4pi, is that correct?</p>
<p>I got 2-pi/4</p>
<p>how did you get 2pi/4 OR pi/2?</p>
<p>no no, I lied, sorry ^_^, it's 8-pi (8 minus pi.)
This is how you do it, if you dont know how to graph it, just plug it in your calculator (in case you dont know how: I'm using a TI-84, go to Mode--> the 4th line, choose "Pol" for Polar coordinates--> then go to y=, and then plug in r=2 and r = 2(1-sinx), then hit graph). Then now you see the region of area that you need to find.
Since the region is symmetrical to the y axis, you can calculate area of the right side, then double it.
Area of the right side: (1/2) integral from 0 to pi/2, of [2^2 - (2(1-sinx))^2] dx. (top squared - bottom squared). Integrate that, you got 8-pi</p>
<p>dont you integrate from 0 to 2pi</p>
<p>no, because the question is "inside r=2 and outside r=2(1-sinx), therefore, if you integrate from 0 to 2pi, the answer will be the area of the two polar function, which is wrong. you can do it from 0 to pi, without doubling it.</p>
<p>also when i do integrate from 0 to pi/2 i get 8+pi</p>
<p>That doesn't sound right van. Justify what you are doing with actual calculus instead of putting what the calculator says.</p>
<p>you know what, I'll double check it >_<</p>
<p>uhmmm I still get 8 - pi</p>
<p>wait van your are correct, it is 8-pi, i made a silly mistake
But please justify why you integrate from 0 to pi/2 , im having trouble finding the boundaries of integration</p>
<p>yes sure, to find the bounds, you set the r's equal to each other. you got theta = 0 and pi, right? you can find the area from 0 to pi :
(1/2) integral from 0 to pi, of [2^2 - (2(1-sinx))^2] dx. (top squared - bottom squared), then u still get 8 - pi.
Another way to do it, because the graph is symmetrical to the y-axis, so u can find the area from 0 to pi/2, then double the answer.
2(1/2) integral from 0 to pi, of [2^2 - (2(1-sinx))^2] dx. (top squared - bottom squared), then u still get 8 - pi.</p>
<p>That's right. My bad.</p>
<p>not a problem! ^_^
.-/-./-.--/-/..../../-./--.//./.-../.../. ??</p>
<p>this is what exactly on my test, same problem, I got it wrong, that's y I remember it lol</p>
<p>thanks van ur a big help :) but wait 1 last question, after i get the r's equal to each other i get 0,pi,2pi, etc.. every pi how did you know to stop at pi?</p>
<p>for these problems, they will tell u an interval, such as 0<theta<2pi (less than or equal to), and you also have to look at the graph. You cant do these problems without graphing it. If you wanna know how to graph it by hands, then lemme know.
Anyother problems?? LOL, I'm a calc lover. ^_^</p>
<p>yea how would you graph this by hand on and xy-plane without using a calculator</p>
<p>well, for the polar-coord problems, you need to graph xy-plane first <of course=""> then draw about 3--> 5 circle with center 0.
With the functions such as r = sin theta, it will be a circle w/ radius 1 lying on y-axis.
With the functions such as r = 2sin theta, it will be a circle w/ radius 2 lying on y-axis and so on.
With the functions such as r = cos theta, it will be a circle w/ radius 1 lying on x-axis.
With the functions such as r = 2cos theta, it will be a circle w/ radius 2 lying on x-axis and so on.
Depend on the function, if it contains sin, then it has to be symmetrical to the y-axis (always), if it contains cos, then it has to be symmetrical to the x-axis (always).
These are just some simple function that you can memorize. For the one which has the angle of 2theta, 4 theta, 6theta... (even numbers), the graph is some kind of a flower w/ 2n number of petals. Example: 2theta will have 4 petals, 4 theta will have 8 petals and so on.
For the one which has the angle of 3theta, 5 theta, 7theta... (odd numbers), the graph is some kind of a flower w/ n number of petals. Example: 3theta will have 3 petals, 5 theta will have 5 petals and so on.
Again, depend on the function os sin or cos, you draw it symmetrical to x- or y-axis.
For other problems that do not involve in these cases, then you need to set up a chart, r and theta (just as x and y), then graph it.</of></p>
<p>This is optional, you dont have to do it by hand you know, cuz you can use a calculator in AP exam. Just because we are not allowed to use a calculator in my class, so that's y I have to learn how to do it. ^_^ have fun</p>