Example 6: Area Under a Curve

(a) Find the area of the region under the curve

for -1 ≤ x ≤ 1.

(b) Find the area of the region under the same curve for -∞ < x < ∞. The regions are shown in Figure 7.3.7.

Figure 7.3.7

(a)

(b)

From the graph of arctan x we see that the first limit is -π/2 and the second limit is π/2, so

Thus the region under y = 1/(1 + x²) has exactly the same area as the unit circle, and half of this area is between x = -1 and x = 1.