## Introduction to Integration - Gaining Geometric Intuition

HELP
`f(x)` =     The integral of `f(x)` from `a` to `b` is .

How should we think of the integral `int_a^b f(x) dx` geometrically? It is basically "area under the curve," but there are some caveats... Some authors call the integral signed area.

In the applet above you can move the blue points on the `x`-axis, and you can even click and drag the function graph around. You can type in your own function in the input box below the applet. Remember, you can Shift + Click and Drag the applet background to move it around.

#### Explore

1. The first and most important thing to do with this applet is play. Take a few minutes and try to get a rough sense of how the integral relates to the shaded area, and how these are affected by the position of `a` and `b`.
2. Reset the applet, and consider its initial settings. The shaded area represents the integral of `-0.05x^2 + 2` from `1` to `2`. Confirm that the shaded area is just under `2`.
3. As `b` is increased from `2` to `6`, the function `f` decreases. Will the integral from `1` to `b` increase or decrease? Make your guess first, then try the applet. Write a sentence or two explaining why the integral behaves the way it does.
4. What "strange" thing happens to the integral when `a = 9` and `b = 10`? Write a rule for how the integral behaves in this situation.
5. Find values for `a` and `b` where `a < b` but the integral of `f(x)` from `a` to `b` is zero. Explain what is happening.
6. Find new values for `a` and `b` so that the integral is not zero. Now what happens if you switch the order of `a` and `b`?
7. Let `f(x) = 2`, make `a = 0` and make `b = 1`.
1. If `b` increases at a steady rate, does the integral increase at a steady rate?
2. Confirm that if `b` decreases and `b` is positive, then the integral decreases. Does this continue to be true even if `b` is negative? Explain why the results you see occur.
8. Let `f(x) = .5 x`, make `a = 0` and make `b = 1`.
1. If `b` increases at a steady rate, the integral does not increase at a steady rate. Descibe what happens, and explain why it happens.
2. Confirm that if `b` decreases and `b` is positive, then the integral decreases. Does this continue to be true even if `b` is negative? Descibe what happens, and explain why it happens.