Line Integrals - Part II — Question 9

Question 9

On the parabola $C:y=x^2$ from $(0,0)$ to $(1,1)$, a constant $a$ satisfies $$\int_C ax\,dx+y\,dy=3.$$ Determine $a$.

Tasks

1. Convert the entire integral to the parameter $x$.

2. Solve the resulting equation for $a$.

3. Substitute the result to verify the specified value.

Question 9 – Solution

Strategy. Use $dy=2x\,dx$ to make the unknown coefficient appear in a simple algebraic equation.

Step 1: Convert Since $y=x^2$ and $dy=2x\,dx$, $$\int_C ax\,dx+y\,dy =\int_0^1\left(ax+x^2(2x)\right)dx.$$ Therefore $$\int_0^1(ax+2x^3)\,dx =\frac a2+\frac 12.$$

Step 2: Solve $$\frac a2+\frac 12=3,\qquad \frac a2=\frac 52,\qquad \boxed{a=5}.$$

Step 3: Verify With $a=5$, $$\int_0^1(5x+2x^3)\,dx =\left[\frac{5x^2}{2}+\frac{x^4}{2}\right]_0^1 =\frac 52+\frac 12=3.$$

Verification The computed value satisfies the original integral exactly. Because the coefficient of $a$ is $\int_0^1x\,dx=1/2\ne 0$, the solution is unique.