[SOLVED] Finding the area inside a loop?

redman223 · #1 $Old$ 11-19-2008, 01:30 PM

The problem says that the curve makes a loop along the x-axis, and asks for the area inside the loop. How do I go about finding this? This seems slightly different than the other area problems we have been given.

x=25-t^2
y=t^3-25t

dx/dt=-2t
dy/dt=3t^2-25
dy/dx=3t^2-25/-2t

But what do I do from here?

redman223 · #2 $Old$ 11-19-2008, 10:05 PM

Anyone know how to solve this problem? I am stuck from this point.

shawsend · #3 $Old$ 11-20-2008, 05:08 AM

Yea I do, I just thought my method was a bit tacky. I mean, how do you solve for the area under a function using the parametric expressions? Is that really easy and I'm suppose to know that already or what (I'm older than you and can't use that excuse)? See what I mean. Ok, so I just wrote it in terms of $y(x)=\pm x\sqrt{25-x}$ . Now look at the plot below:

For large negative t, both x and y are negative. When t hits -5, they're zero, into the first quad, looping down, reaching the x-axis when t=0 right? Then it goes into the 4th quad, back to the origin at t=5 and then into the 2nd quad. So I'd solve for y in terms of x:

$y(x)=\pm x\sqrt{25-x}$

We can integrate just the top part since it's reflective and I'll use the positive square root:

$2\int_0^{25} x\sqrt{25-x}dx$