In this lesson you will approximate the area of a region that is bounded by a curve by using rectangles whose heights are determined by the y-coordinates of points on the curve. You can use rectangles to the right of the curve, to the left of the curve, or centered on the curve. These are called Riemann Sums and allow you to approximate the area under the curve. This applet lets you compare the different approximation methods simultaneously along with comparing the area varying numbers of rectangles.[br][br]To change the number of rectangles used to approximate the area, use the slider and increase/decrease "n". You can also change which sum is visible by checking/unchecking the box next to the desired approximation method.

1. How does the Left Riemann Sum compare to the Right Riemann Sum? Why?[br][br]2. Which approximation is closest to the actual area under the curve?[br][br]3. Determine the area under the curve and above the x-axis for the function f(x) = -x^2 + 4 on the interval [0, 2]. Use all three approximation methods.

To approximate the area under a graph we split the region up into many thin rectangles. Drag the slider labeled n to increase the number of rectangles used.[br][br]The height of the approximating rectangle can be taken to be the height of the left endpoint, the height of the right endpoint, or the height of any point in the subinterval. Drag the slider labeled "position" to change the point used for the height of the rectangle.[br][br]The sum of the areas of all the thin rectangles is the Riemann Sum displayed.[br][br]If you wish to change the function f, say to sin(x), then just type f(x)=sin(x) in the input field at the bottom of the applet.