Line Integrals - Part II — Question 3

Question 3 – Solution

Strategy. A combined differential-form integral becomes one ordinary integral after substituting $dx=x'(t)dt$ and $dy=y'(t)dt$.

Step 1: Parameter data $$\mathbf r(t)=\langle 2t,t\rangle,\qquad 0\le t\le 1,$$ so $$x+y=3t,\quad x-y=t,\quad dx=2\,dt,\quad dy=dt.$$

Step 2: Evaluate $$\begin{align*} \int_C (x+y)\,dx+(x-y)\,dy &=\int_0^1\big[(3t)(2)+(t)(1)\big]\,dt\\ &=\int_0^1 7t\,dt =\boxed{\frac 72}. \end{align*}$$

Verification At $t=0$ the parametrization gives $A$, and at $t=1$ it gives $B$, so the orientation is correct. Directly grouping the transformed integrand gives $[2(x+y)+(x-y)]dt=(3x+y)dt=7t\,dt$, confirming the algebra.