Line Integrals - Part II — Question 7

Question 7 – Solution

Strategy. Parametrize by the changing coordinate $y$; the constant coordinate has zero differential.

Step 1: Parametrize $$\mathbf r(t)=\langle 2,t\rangle,\qquad 0\le t\le 3.$$ Thus $dx=0$, $dy=dt$, and $x+y=2+t$.

Step 2: Evaluate $$\boxed{\int_C(x+y)\,dx=0}$$ because $dx=0$ everywhere. Meanwhile, $$\int_C(x+y)\,dy =\int_0^3(2+t)\,dt =\left[2t+\frac{t^2}{2}\right]_0^3 =\boxed{\frac{21}{2}}.$$

Step 3: Representation issue A vertical segment assigns many $y$-values to the single value $x=2$, so it cannot be written as a single-valued graph $y=g(x)$. Parametrization avoids this limitation.

Verification The second integrand rises linearly from $2$ to $5$; its average $7/2$ times the $y$-interval length $3$ gives $21/2$.