Line Integrals - Part II — Question 10

Question 10 – Solution

Strategy. Use the shared coordinate parameter $x$ on both paths, then compare the scalar functions traced between the same endpoints.

Step 1: Straight path On $C_1$, $y=x$, so $$\int_{C_1}y\,dx =\int_0^1x\,dx =\boxed{\frac 12}.$$

Step 2: Parabolic path On $C_2$, $y=x^2$, so $$\int_{C_2}y\,dx =\int_0^1x^2\,dx =\boxed{\frac 13}.$$

Step 3: Interpretation Both curves have the same initial point, terminal point, and orientation, yet $$\frac 12\ne\frac 13.$$ Therefore the value of $\int_Cy\,dx$ is not determined by endpoints alone; it depends on the path between them.

Verification For $0<x<1$, the line $y=x$ lies above the parabola $y=x^2$. With $dx>0$, its integral must be larger, consistent with $1/2>1/3$.