Using Integration to find the Area between 2 Curves

Using Integration to find the Area between 2 Curves

What if you want to determine the area between two curves, y2and y1 , like the area shaded below?

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

   dA = (y2-y1) dx

Where y2 is the top curve and y1 is the bottom curve.

Note : Sometimes using dA = (y2-y1) dx won’t work, but dA = (x21 – x1) 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 y2= 2x – x2 and y1= x4.

NEED SKETCH

   dA = (y2-y1) dx = ((2x – x2 )-  x4) dx

Solve for the limits:

2x – x2 = x4

 x4+ x2 – 2x = 0

x(x3+ x – 2) = 0

x = 0, 1    So:

       

            A = x2 – x3/3 – x5/5 between 0 and 1

        A = 7/15

Practice Problems