# Fundamental theorem of calculus exploration

- Author:
- Chris, Tom Owsiak

- Topic:
- Calculus, Integral Calculus

Riemann sums and the Fundamental Theorem of Calculus

Answer the questions below by using the graph above. You will need to enter the new function in the box above and may have to grab the graph and lift it to move the dots on the x-axis to x=1 and x=5.

1. For the function above on the interval [1,5], select n=4.
(a) Is the left or right sum an upper bound for ?
(b) For these Riemann sums, what is a bound on the integral?
(c) Compute the Riemann sum for this case instead using the **midpoint** of the interval as in our text.

2. Use the same function and interval as in question #1. (a) As you slide the values of n to be larger, what is true about the left and right Riemann sums? (b) When you slide n to its largest value, how accurately can you determine the definite integral? Be specific.

3. Use the fundamental theorem of calculus to explicitly solve Show all your steps.