Approximating a Definite Integral from Graph
- Dr. Jack L. Jackson II
- Definite Integral
Approximation Technique and Instructions
We are trying to approximate from the graph. Setup In the app enter a formula for a function, f(x), in the input box. Adjust values of the limits of integration a and b via the sliders or input boxes. This app assumes -10 a b 10. Visualize the Area Select the checkbox for Definite Integral Area to see the related area between the graph of f and the x-axis on the interval [a, b]. The definite integral is the area of the shaded portion above the x-axis minus the area of the shaded portion below the x-axis. Subdivide We will approximate this by a Riemann Sum where we draw in rectangles of width 1/2. Select the checkbox for vertical subdivision to see this subdivision. Draw Rectangles There are points which start on the x-axis at x-values which are at multiples of 1/2. If you move one of these points you will produce a rectangle of width 1/2 with one base on the x-axis. Adjust the height of appropriate rectangles to the average height of the function on that subinterval. If you do this correctly, then the amount of area that is part of the integral area that is sticking out of the rectangle should be the same size as the area that is inside the rectangle but beyond the function. Repeat for all rectangles over the interval [a, b]. Compute the Riemann Sum Estimate the signed heights of these rectangles from the graph. Don't forget that this is negative for rectangles going below the x-axis, and it is positive for rectangles going above the x-axis. Add these heights. Multiple the sum by the common width. This will produce a Riemann Sum which is approximately the same as the area of your rectangles. Check Your Work Activate the Average Value Rectangles checkbox to see the best place to draw the rectangles. Did you at least get pretty close or were you off a bit on some? Activate the Definite Integral Value checkbox to see a 5 decimal place value of the definite integral. How close was your answer? Repeat for different intervals and different functions for more practice.