Labware - MA35 Multivariable Calculus - Single Variable Calculus



Riemann Integral


We can approximate the area underneath the graph of a function with a sequence of rectangles.

Consider a function f(x) and some interval a ≤ x ≤ b. To approximate the area underneath the graph of f(x) over this interval, start by forming a partition of [a,b] into n segments. That is choose points \{x1, x2, ..., xn-1\} such that a = x0 < x1 < ... < xn-1 < xn = b. Then, construct a rectangle for each segment [xj, xj+1] such that the side of the rectangle opposite this segment intersects the graph of f(x). The sum of the areas of this sequence of n rectangles is called a Riemann sum.



  • 1. Compare (using <, >, =) the values of the Riemann Integral for -11x2dx for:
    • 1 subdivision
    • 5 subdivisions
    • 20 subdivisions
  • 2. Compute the Riemann integral for 01x3dx for three subdivisions, and use the lab to verify that your result is reasonable.