Question 27.4: Evaluate the integral ∫C 5y² dx + + 2xy dy along the curve, ......
Evaluate the integral \int_C 5 y^2 \mathrm{~d} x+2 x y dy along the curve, C, consisting of the x axis between x = 0 and x = I and the line x = 1 as shown in Figure 27.6.
We evaluate this integral in two parts because the curve C is made up of two pieces. The first piece C_1 is horizontal, and the second C_2 is vertical. The required integral is the sum ofthe two separate ones. Along the x axis, y = 0 and dy = 0. This means that both the terms 5y² dx and 2xy dy are zero, and so the integral reduces to
\int_{x=0}^{x=1} 0 \mathrm{~d} x=0and so there is zero contribution to the final answer from this part of the curve C. Along the line x = 1, the quantity dx equals zero. Hence 5y² dx also equals zero, and 2xy dy equals 2y dy. Because y ranges from 0 to 1 the integral becomes
\int_{y=0}^{y=1} 2 y \mathrm{~d} y=\left[y^2\right]_0^1=1Note that this is a different answer from that obtained in Example 27.3.