Area between Curves

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 = x² from x = 0 to x = 1.
	\(\eqalign{{\rm{Area}} &= \int_0^1 {\left\| {x - {x^2}} \right\|dx} \\ &= \int_0^1 {\left( {x - {x^2}} \right)dx} \\ &= \left. {\left( {\frac{1}{2}{x^2} - \frac{1}{3}{x^3}} \right)} \right\|_0^1\\ &= \left( {\frac{1}{2} - \frac{1}{3}} \right) - \left( {0 - 0} \right)\\ &= \frac{1}{6}}\)
Example 2:¹	Find the area between x = y + 3 and x = y² from y = –1 to y = 1.
	\(\eqalign{{\rm{Area}} &= \int_{ - 1}^1 {\left\| {y + 3 - {y^2}} \right\|dy} \\ &= \int_{ - 1}^1 {\left( {y + 3 - {y^2}} \right)dy} \\ &= \left. {\left( {\frac{1}{2}{y^2} + 3y - \frac{1}{3}{x^3}} \right)} \right\|_{ - 1}^1\\ &= \left( {\frac{1}{2} + 3 - \frac{1}{3}} \right) - \left( {\frac{1}{2} - 3 + \frac{1}{3}} \right)\\ &= \frac{{16}}{3}}\)