Improper Integral with Diverging Integrand
This applet illustrates the difficulties with divergence of the integrand in the interior of the integration interval.
- The red diamond traces the total area enclosed between the curve and the axis.
- The (symmetric) limits of integration are set by moving point C.
- Drag point A to simulate taking the limit approaching the origin from left and right.
- The sliders n and m control the number of rectangles in the Riemann's sums representing the integrals on the right and left respectively of the divergence.
- The sliders Rpos and Lpos control where in each interval the height of the rectangles for the Riemann's sums is set.
- The field labeled f(x) permits other functions to be entered. NOTE: only divergence at x=0 will be properly dealt with
It is possible for f(x)=1/x to have a non-divergent answer for special cases of values of n, m Lpos, Rpos, and LRratio. However, the requirement for the integral to be defined is that we get the SAME answer no matter how we construct the Riemann's sums in for the two pieces of the integral.