Example 1: Horizontal and Vertical Line

EXAMPLE 1

Find the line integral of

F(x, y) = sin x cos yi + e^xyj

along

(a)    The horizontal line C₁: 0 ≤ x ≤ π, y = π/3 (Figure 13.2.7(a)).

(b)    The vertical line C₂: 0 ≤ y ≤ 1, x = 2 (Figure 13.2.7(b)).

Figure 13.2.7

We use partial integrals, (a)

(b)

Given two points A and B, there are infinitely many different smooth curves C from A to B. In general the value of a line integral will be different for different curves from A to B.