**Prerequisites:** Riemann Sums, Derivative

**Goal:** To understand the definite integral function

- 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? - What happened to the yellow curve as you performed step 1?
- Move the top slider further to the right. How does the total area between the curve and the
*x*-axis change? - What happened to the yellow curve as you performed step 3?
- 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)? - 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)? - Can area ever be negative?
- What does the number at the right endpoint of the yellow curve represent?
- What does the yellow curve represent?
- What is the relation between the integral of
*f(x)*and the shaded area?