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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.

 

 

\(\eqalign{{\rm{Area}} &= \int_{ - 1}^1 {\left| {y + 3 - {y^2}} \right|dy} \\ &= \int_{ - 1}^1 {\left( {y + 3 - {y^2}} \right)dy} \\ &= \left. {\left( {\frac{1}{2}{y^2} + 3y - \frac{1}{3}{x^3}} \right)} \right|_{ - 1}^1\\ &= \left( {\frac{1}{2} + 3 - \frac{1}{3}} \right) - \left( {\frac{1}{2} - 3 + \frac{1}{3}} \right)\\ &= \frac{{16}}{3}}\)

 

See also

Area under a curve, definite integral, absolute value rules

 


  this page updated 19-jul-17
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
Copyright © 2000 by Bruce Simmons
All rights reserved
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