Using Integration to find the Area between 2 Curves

What if you want to determine the area between two curves, y_{2}and y_{1} , like the area shaded below?

We use the same approach: sketch the differential area dA, then write the equation:

dA = (y_{2}-y_{1}) dx

Where y_{2} is the top curve and y_{1} is the bottom curve.

Note : Sometimes using dA = (y_{2}-y_{1}) dx won’t work, but dA = (x_{21} – x_{1}) dy will.

Then you need to find out where the two curves intersect. Set the 2 equations equal and solve for the x. You should get two solutions. These are the limits of integration.

Example: Find the area between y_{2}= 2x – x^{2} and y_{1}= x^{4}.

NEED SKETCH

dA = (y_{2}-y_{1}) dx = ((2x – x^{2} )- x^{4}) dx

Solve for the limits:

2x – x^{2} = x^{4}

x^{4}+ x^{2 }– 2x ^{ }= 0

x(x^{3}+ x – 2) = 0

x = 0, 1 So:

A = x^{2} – x^{3}/3 – x^{5}/5 between 0 and 1

A = 7/15