Intuition 1:
1. Notice that the area under the curve from 0 to π is 2.
2. Drag b (slider or point) to 2π. Notice that the integral is 0.
3. Drag a (slider or point) to π. Notice that the integral is -2.
⇒ this brings out the fact that if the curve is under the x axis, then the integral is negative.
If you want to find the area between the curve sin(x) and the x-axis from 0 to 2π, you have to be ware that first find the area between 0 and π (= 2), and then add this to the magnitude of the area from π to 2π (= |-2|).
i.e.
Intuition 2:
1. Now try dragging b (slider or point) to 0 (and a remains in π). Notice that the integral is -2.
2. Invert a and b. Notice that the integral becomes 2.
3. Drag b to -π and a to 0. Notice that now the integral is 2.
4. Invert a and b. Notice that the integral becomes -2.
⇒ this brings out the fact that if you invert the limits a and bin an integral, the sign of the integral gets inverted.
i.e.