Area between Curves
The
area between curves is given by the formulas below.
Formula 1: 
Area = \(\int_a^b {\,\,\left {f\left( x \right)  g\left( x \right)} \right\,\,\,dx} \)


for a region bounded above by y = f(x) and below by y = g(x), and on the left and right by x = a and x = b.

Formula 2: 
\(\int_c^d {\,\,\left {f\left( y \right)  g\left( y \right)} \right\,\,\,dy} \)


for a region bounded on the left by x = f(y) and on the right by x = g(y), and above and below by y = c and y = d.

Example 1:^{1} 
Find the area between y = x and y = x^{2} from x = 0 to x = 1.


\(\eqalign{{\rm{Area}} &= \int_0^1 {\left {x  {x^2}} \rightdx} \\ &= \int_0^1 {\left( {x  {x^2}} \right)dx} \\ &= \left. {\left( {\frac{1}{2}{x^2}  \frac{1}{3}{x^3}} \right)} \right_0^1\\ &= \left( {\frac{1}{2}  \frac{1}{3}} \right)  \left( {0  0} \right)\\ &= \frac{1}{6}}\) 
Example 2:^{1} 
Find the area between x = y + 3 and x = y^{2} from y = –1 to y = 1.


\(\eqalign{{\rm{Area}} &= \int_{  1}^1 {\left {y + 3  {y^2}} \rightdy} \\ &= \int_{  1}^1 {\left( {y + 3  {y^2}} \right)dy} \\ &= \left. {\left( {\frac{1}{2}{y^2} + 3y  \frac{1}{3}{x^3}} \right)} \right_{  1}^1\\ &= \left( {\frac{1}{2} + 3  \frac{1}{3}} \right)  \left( {\frac{1}{2}  3 + \frac{1}{3}} \right)\\ &= \frac{{16}}{3}}\) 
See
also
Area
under a curve, definite
integral, absolute value rules
