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. 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? 2. Which approximation is closest to the actual area under the curve? 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.