# Calculus Magic "Trapped" in Riemann Sums

## Discovery Activity #1

## 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.....

## 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?

## Question 2

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

## 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

## 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!

## Question 8

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