# CLEAR Calculus

## Lab 16: The fundamental theorem of calculus - part 2

1. Let $$F$$ be an antiderivative for $$f$$. Explain what this means, both algebraically and graphically.

2. Draw a graph in which the definite integral $$\int_a^b f(x)\ dx$$ is approximated by a Riemann sum using an arbitrary evaluation point for each subinterval and write out the Riemann sum. Shade in the region that represents the error for this approximation (accounting for positive and negative contributions). Explain how this error can be made smaller than any desired bound.

3. State the Mean Value Theorem applied to the function $$F$$ over a single subinterval from your algebra and graph in Question 2. Draw a picture of a single subinterval to illustrate this.

4. Use the special input values for $$f$$ obtained from Question 3 as the evaluation points for each subinterval in your expression from Question 2. Draw a picture to illustrate this.

5. What happens to the Riemann sum? What happens to the error? What can you conclude about the integral?