How do I Calculate the Area Under a Curve Using Precalculus?

Graph of function on a grid.

Pre-calculus is the course that precedes calculus and addresses advanced algebraic techniques in order to familiarize the student with the concepts that led to the discovery of calculus. While the most accurate way to calculate the area enclosed by a curve is to perform an operation in calculus called integration, a lesson on how to approach this type of problem using methods in pre-calculus will give the student a good introduction to the logic behind the methods used in calculus.

Instructions

1. Graph the function for the curve. The function will take the form of y = f(x). This can be done by hand, on a computer or on a graphing calculator. However you choose to do this, ensure that you are able to print the graph, or accurately draw the curve displayed by hand. You will want to be able to inspect the graph closely and draw on it for a visual reference.

2. Choose the x-range over which the area is to be calculated. Draw a line parallel to the x-axis at the lowest y-value on the portion of the curve that falls within the x-range. This will give you a rectangle whose area can be calculated using the length and width of the shape as read from the axes.

3. Choose a small number to act as an increment in the x-direction. The increment only need be small in comparison to the range in x that is covered by the problem. If the problem is given in a textbook, the problem statement will likely tell you what the increment should be.

4. Draw smaller rectangles whose upper left corners touch the y=f(x) curve with widths in the x-direction that are equal to the increment chosen in Step 3. The heights of the rectangles will vary with the height of the curve. Your graph should now show a large rectangular base topped by skinnier rectangles that trace out the area underneath the curve.

5. Calculate the area for every rectangle that appears in the graph. Add all of the areas. This will give you an under-approximation---only slightly smaller than the real value---of the area enclosed by the graph. The entire procedure can be repeated with the upper right corner of the triangles touching the curve, resulting in an over-estimate of the area, for a more thorough analysis.

Tips & Warnings

• This procedure is the basis for the calculus integration method used for these types of problems. Integration will yield the sum of the rectangles as the increment in Step 3 becomes infinitesimally small, yielding the precise value of the area enclosed by the curve.

References

• Precalculus
• Integrated Publishing - Math: Area Under a Curve