how to do this problem?
Use a line integral to find the area of the regionR bounded by the graphs of y = 2x + 1 and
y = 4 − x^2
Green's theorem says that
$\displaystyle \oint f(x,y)dx+ g(x,y)dy$$\displaystyle = \int\int \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right) dx dy$
Since the area of a region is just $\displaystyle \int\int dx dy$, choosing any f and g such that $\displaystyle \frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}= 1$ will convert that double integral for area into a line integral over the boundary. A simple and valid choice is g(x,y)= x, f(x,y)= 0. The area is given by $\displaystyle \int x dy$ .
You will have to convert both parabola and line to parametric equations. Be careful about integrating in the right direction- you must integrate counter-clockwise around the boundary.
Frankly, this is much harder than just integrating the difference of the two functions!
Show us what you have tried. Did you write parametric equations for the line and parabola? It's straight forward just to use x itself as parameter. On the line, y= 2x+1, dy= 2dx and on the parabola, $\displaystyle y= 4- x^2$, dy= -2xdx[/tex]. At what points do the line and parabola intersect?
Actually, looking at this more closely, I take back what I said before- the integrals are not difficult at all!
Using LaTex. You can click on the "sigma" button third from the right just above the window you are typing in or just type [ math][/ math] (without the spaces I had to put in so you would be able to see those) yourself and put the code between them. There is a LaTex tutorial thread at http://www.mathhelpforum.com/math-help/latex-help/ but you can also see the code used for a particular formula by clicking on the formula.