Integral of a Straight Line

Given b > 0, evaluate the integral

The area under the line y = x is divided into vertical strips of width dx. Study Figure 4.1.15.

Figure 4.1.15

The area of the lower region A is the infinite Riemann sum

(1)

area of

By symmetry, the upper region B has the same area as A;

(2)

area of A = area of B.

Call the remaining region C, formed by the infinitesimal squares along the diagonal. Thus

(3)

area of A + area of B + area of C = b².

Each square in C has height dx except the last one, which may be smaller, and the widths add up to b, so

(4)

0 ≤ area of C ≤ b dx.

Putting (1)-(4) together,

Since b dx is infinitesimal,

Taking standard parts, we have