Prerequisites: Riemann Sums, Derivative

Goal: To understand the definite integral function

1. The blue curve in the Java applet represents the graph of an arbitrary function, y=f(x). Move the top and bottom sliders so that both endpoints are at -3.0. Then move the top slider slowly to the right as far as -1.5. How does the total area between the curve and the x-axis change?
2. What happened to the yellow curve as you performed step 1?
3. Move the top slider further to the right. How does the total area between the curve and the x-axis change?
4. What happened to the yellow curve as you performed step 3?
5. Place the bottom slider at -3.0 and the top slider at -0.33. The right endpoint of the yellow curve now has a y-value of (close to) zero. What do you notice about the shaded area(s)?
6. Place the bottom slider at -3.0 and the top slider at 0. The right endpoint of the yellow curve now has a negative y-value. What do you notice about the shaded area(s)?
7. Can area ever be negative?
8. What does the number at the right endpoint of the yellow curve represent?
9. What does the yellow curve represent?
10. What is the relation between the integral of f(x) and the shaded area?