Area Between 2 Curves (DP IB Analysis & Approaches (AA): SL): Revision Note

Area between 2 curves

What do we mean by 'area between two curves'?

• Areas whose boundaries include two curves can be found by integration

  • The area between two curves will be the difference of the areas under the two curves

    • both areas will require a definite integral

  • Finding points of intersection may involve a more awkward equation than solving for a curve and a line

How do I find the area between two curves?

• STEP 1

  If not given, sketch the graphs of both curves on the same diagram

  • You can use a GDC to help with this step
     

•  STEP 2

  Find the intersections of the two curves

  • If no diagram is given this will help identify the area(s) to be found

• STEP 3

  For each area (there may only be one) determine which curve is the ‘upper’ boundary

  For each area, write a definite integral of the form${\int }_a^b\left(y_1-y_2\right)\mathrm{d}x$

  • where$y_1$ is the function for the ‘upper’ boundary and$y_2$ is the function for the ‘lower’ boundary

•  STEP 4

  Evaluate the definite integrals and sum them up to find the total area