Calculus Magic "Trapped" in Riemann Sums

Author:
pbattaglia
Topic:
Calculus

Discovery Activity #1

The graph of is shown below. We are going to be looking at the interval [0.5, 2]. Follow the prompts for the activity.

Lower Sum Approximation

Check the "Lower Sum" box and then enter the approximate area given by the lower sum.

Upper Sum Approximation

Now uncheck the "Lower Sum" box and check the "Upper Sum" box. Enter the approximate area given by the upper sum.

Trapezoidal Sum Approximation

Now uncheck the "Upper Sum" box and check the "Trapezoidal Approximation" box. Enter the approximate area given by the trapezoidal sum.

Before answering the next few questions.....

Change the function given and see if you can nail down a generalization as to what graphical feature causes a lower/upper sum to be an over/underestimate as compared to the true area under the curve.

Question 1

Adjust the slider to . Check only the "Lower Sum" box. It appears that the lower sum will yield which type of area approximation?

Check all that apply

Question 2

Now turn the "Lower Sum" off and turn the "Upper Sum" on. The upper sum will yield which type of area approximation?

Check all that apply

Question 3

Fill in the blank: If a function is __________________, then a lower sum will yield an overestimate of the actual area under the curve and an upper sum will yield an underestimate of the actual area.

Question 4

Fill in the blank: If a function is __________________, then a lower sum will yield an underestimate of the actual area under the curve and an upper sum will yield an overestimate of the actual area.

Discovery Activity #2

The graph below shows only the trapezoidal sum being used to find the area under a curve. As you play around with this graph, feel free to move the slider or the points. You can even input a different function if you like. What are you looking for? Good question! You'll notice that at some points, the trapezoids are ABOVE the curve, and at other points, the trapezoids lie BELOW the curve. What feature of the graph affects whether a trapezoidal sum is an over/underestimate of the true area under the curve? When you feel confident, answer the questions that follow the graph.

Question 6

If the graph of a curve is ____________, then a trapezoidal sum will yield an underestimate as compared to the true area under the curve.

Question 7

If the graph of a curve is ____________, then a trapezoidal sum will yield an overestimate as compared to the true area under the curve.

The BIG REVEAL!

What if I told you that there actually is a connection between lower and upper sums, and the trapezoidal sum? For real! Look at the graph we used in Activity #1. Pick a certain number of rectangles and take note of the lower and upper sums. Then take note of the trapezoidal sum. Notice anything? Keep looking until you find it!

Question 8

The trapezoidal sum just happens to be the __________ of the lower and upper sums!