Question

Can someone please explain the idea behind the process of finding the area under a curve given a pair of parametric equations?

Answer 1

(Please answer using Calculus techniques)

Answer 2

ok, we know that the area under a curve y=F(x) from a to b is:\[A=\int\limits_{a}^{b}F(x)dx\]where \[F(x)\ge0\]If the curve is given by parametric equations x=f(t) and y=g(t), where t is between two values c and d and x(c)=a, x(d)=b. We can adapt the usual substitution rule to evaluate these as follows:\[A=\int\limits_{a}^{b}y dx=\int\limits_{c}^{d}g(t)f'(t)dt\]because y=g(t) and f'(t)=dx/dt, and we make sure to change our limits of integration for our new variable "t". Now the reason we use (y)dx in the integral above can be understood by breaking up your curve into rectangles with height y and width dx. Our area is the limit of the sum of these rectangles as usual.

Answer 3

so if given the bounds of integration in the problem, plug those values into the x equation (if finding the area of a curve around the x-axis, when y = 0)?

Answer 4

we don't care about y when setting the bounds of integration. We just need to know our start point and end point. Now when we switch from y and dx to the parametric equations we switch the bounds from x=a and x=b. We need to solve x(t)=a and x(t)=b for the values of t at which our function equals a and b, respectively. I've called these t=c and t=d.