Approximating the Definite Integral Using Midpoints
by Mike Martin

On this page, you can vary the number of partitions, N, used to approximate the definite integral for

on the interval a < x < b using midpoints on each subinterval. If the function is non-negative on [a, b], then the definite integral on [a, b] can be interpreted as the area of a region R (the region R being bounded above by the graph of y = f ( x ), below by the x-axis, and vertically by the lines x = a and x = b). Submit real values for all of the parameters to obtain a numeric approximation of the definite integral (area), the error involved, and a visualization of the region with the rectangles used. Note that in on this page the concept of ∞ should be investigated using just a relatively large number; also, since there is a singularity in the integrand at x = 0 you should use an integration bound near zero (but not equal) to investigate. On this page, the height of the rectangles is determined by utilizing the x-value of the midpoint of each subinterval.

Enter in the value of the parameters and select the plotting bounds, then press the "Plot/Evaluate" button.

k =  

p =  

a =  

b =  

N =  

The approximation is

The error in the approximation is


Wikipedia's Definite Integrals

Wikipedia's Improper Integrals

Wikipedia's Numerical Integration

MathWorld's Integral

Wolfram's Integration