Example 5: Area Under a Line

Find the area under the line y = 1 + 3x from x = 0 to x = 1.

This can be done either with or without a change of variables.

Without change of variable: (1 + 3x) dx = x + 3x²/2 + C, so

With change of variable: Let u = 1 + 3x. Then

du = 3 dx, dx = 1/3du.

When

x = 0, u = 1 + 3 · 0 = 1.

When

x = 1, u = 1 + 3 · 1 = 4.

Figure 4.4.2