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