Use the input box to define the function f(x). [br][list][*]Click "Trace Area" and use the slider tool for c to trace out the area under the curve. When you drag c all the way to the right endpoint it will reveal the actual area under the graph expressed in integral notation. [/*][*]Check the Rectangles and Partition boxes to reveal rectangles that can be used to estimate this area. [/*][*]The "Rectangles Height" slider determines how the heights of the rectangles are determined. The default is the Midpoint Rule (i.e., the rectangle height is determined by the value of the function at the midpoint of each subinterval), but you can slide to see the Left Sum or Right Sum. [/*][*]The partition slider determines how many rectangles n, which simultaneously determines the widths of the rectangles. [/*][/list]

To estimate the area under the graph of a function over an interval [a, b] using rectangles:[br][list][*]Partition the interval [a, b] into n sub-intervals of length [math]\Delta x=(b-a)/n[/math]. This fixes the width of the rectangles and creates sub-intervals [math][x_0,x_1],[x_1,x_2],\ldots,[x_{n-1},x_n][/math]. [/*][*]Choose a method for determining the height of the rectangles using the value of the function at some point in the sub-interval. For the midpoint rule, for example, choose the midpoint [math]x_i^*[/math] of the sub-interval [math][x_{i-1},x_i][/math] and evaluate [math]f\left(x_i^*\right)[/math] to be the height of rectangle [math]i[/math]. [/*][/list]Calculate the area of each rectangle by multiplying the height [math]f\left(x_i^*\right)[/math] by the width [math]\Delta x[/math], and add up all the areas:[br][list][*][math]f\left(x_1^*\right)\Delta x+f\left(x_2^*\right)\Delta x+\cdots+f\left(x_n^*\right)\Delta x[/math] [br][br][math]=\sum_{i=1}^nf\left(x_i^*\right)\Delta x[/math][/*][/list]