Question

How to find the area of the restricted area in the plane enclosing the curve
r(t) ... (attachment)

Apparently it will look like a ... bowtie; however I do not know how I should proceed in order to ... integral it. Step by step instruction would be highly appreciated :)

Answer 1

If it was a normal integral, I would not have any problem; but I am not sure how to ... handle it in that form (i, j)

Answer 2

If the question is too vague; let me know.
It's basically "find the area" - which is just to use integral on it - however I am not sure how to make it ready for .. integral...ing.

Answer 3

the simplest approach is set out here:
http://tutorial.math.lamar.edu/Classes/CalcII/ParaArea.aspx

and your curve looks like this:
https://www.desmos.com/calculator/nuzxiwcn53

2pi is one full oscillation but you can also see symmetry so you could 4 * one quarter.

Answer 4

I was actually looking at Paul's - however when I try to find a and b, I get 0 on both of them (sin(0) and sin(2pi) = 0)

I can of course "see" that the range is from 0 to 1, both in x and y direction - but I need to figure it out mathematically :)

Answer 5

using Paul's notation \(\alpha = 0, \beta = 2 \pi\) but you may wish to use \(4* \int \limits_{0}^{\pi/2}\) to get round the fact that the negative areas will give you a nil answer overall if you go \( \int \limits_{0}^{2\pi}\).

Answer 6

Okay. I think I got most of it. However, I am unable to figure out how I would figure out that I have to add 4* from 0 to pi/2. I know I can *See* it, using a calculator; It is obvious I can do it - but what if I do not have a graphic calculator? Basically; how do you know you have to change from 0 to 2pi to 0 to pi/2?

Answer 7

Then again; I guess this is a task that require you to use a calculator...

Answer 8

Thanks for the help - got 8/3 as an answer :)