Area of region example

Find area of region bounded by parabola x = y2 and the line y = x.

Graph:

\includegraphics[width=2.5in]{intparabline.eps}

We'll let y go from 0 to 1. Then x goes from y2 to y.

Area is

$\displaystyle \int_{0}^{1}$$\displaystyle \int_{{y^2}}^{y}$dx dy = $\displaystyle \int_{0}^{1}$$\displaystyle \biggl($x$\displaystyle \bigg\vert _{{x=y^2}}^{{x=y}}$$\displaystyle \biggr)$dy    
  = $\displaystyle \int_{0}^{1}$(y - y2)dy    
  = $\displaystyle \biggl[$$\displaystyle {\frac{{y^2}}{{2}}}$ - $\displaystyle {\frac{{y^3}}{{3}}}$$\displaystyle \biggr]_{0}^{1}$    
  = $\displaystyle {\frac{{{1}}}{{2}}}$ - $\displaystyle {\frac{{{1}}}{{3}}}$ = $\displaystyle {\frac{{{1}}}{{6}}}$    

Note: if asked to integrate a function f (x, y) over that region, you'd need to compute

$\displaystyle \int_{0}^{1}$$\displaystyle \int_{{y^2}}^{y}$f (x, y)dx dy.    


Duane Nykamp
nykamp@math.umn.edu
2005-09-28