Area Between a Curve and the x-axis

The purpose of this exercise is to deepen understanding of the Riemann integral and its properties by using elementary geometry to calculate the value of the integral for a piece-wise defined function consisting of segments and a semicircle.

Click here to display the area applet in a separate window.      Smaller applet

This is the graph of  

1.

Drag point a to 4, the extreme leftmost point in the domain of f.  Drag point b as close to −3 as possible then click the Show Region button.  Why is the value of this signed area 1?

It may be easier to accurately position points if you enlarge the graph by moving the unit point (1, 0) further away from the origin. 

2.

Drag point b as close to 2 as possible.  Why is the area of the signed region zero instead of the sum of the areas of the two triangles?

3.

Drag point b as close as possible to the origin at 0.  How do you use elementary geometry and integral properties to calculate the area of this region?

4.

Position a at the origin and b at 4.  What is the exact value of ?  What is the 3 decimal point approximation to this value?  Does the applet agree?

5.

Position point b at 4 and a at 3.  Why is the area of this region  positive although the region lies below the axis?

6.

Position point b at the origin and a at 4.  Why is the area of this region  negative although the region lies above the axis?

7.

What should be the exact value of ?  Explain this using elementary geometry and integral properties.