Use the following applets to visualize the Riemann sums. In each, as you increase the nRectangles slider, you will only be able to modify the most recently added rectangle, so you will have to increase it one number at a time to build the rectangles until you have reach the correct number. Set each rectangle by moving the corners so that one corner that has a point on it is on the graph at the left (or right) side of the interval, while the opposite corner is on the axis at the other end of the interval.

Use this applet to represent the results of part a, with a left-hand Riemann sum with $5$ intervals.

**Feedback from applet**

number of rectangles:

rectangles:

Use this applet to represent the results of part b, with a right-hand Riemann sum with $5$ intervals.

**Feedback from applet**

number of rectangles:

rectangles:

Use this applet to represent the results of part c, with a left-hand Riemann sum with $10$ intervals.

**Feedback from applet**

number of rectangles:

rectangles:

Why are the estimates with $5$ intervals so similar?

Even though the two estimates with $5$ intervals give similar answers, we can see from the applets that the estimate with $10$ intervals captures the full height of the graph, which is missed by the sums with $5$ intervals. It is a much better estimate, although smaller intervals would still improve the estimate.

##### Hint

In order for the applet to to recognize the rectangle is the correct height, the upper point on the rectangle must be on the function. You can drag the points across from each other so that the upper point is either the left edge or the right edge of the rectangle.

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