Line Integrals - Part I — Question 1

Question 1 – Solution

Strategy. Use linear interpolation between the endpoints, then replace $ds$ by $|\mathbf r'(t)|\,dt$.

Step 1: Parametrize the segment $$\mathbf r(t)=\langle 3t,4t\rangle,\qquad 0\le t\le 1.$$

Along the curve, $$x+y=3t+4t=7t.$$

Step 2: Arc-length element $$\mathbf r'(t)=\langle 3,4\rangle,\qquad |\mathbf r'(t)|=5,\qquad ds=5\,dt.$$ Therefore $$\int_C(x+y)\,ds =\int_0^1(7t)(5)\,dt =35\left[\frac{t^2}{2}\right]_0^1 =\boxed{\frac{35}{2}}.$$

Verification The speed is constant and the parameter interval has length $1$, so the curve length is $5$, agreeing with the distance formula $\sqrt{3^2+4^2}=5$. The integrand rises linearly from $0$ to $7$, whose average is $7/2$; multiplying by length $5$ again gives $35/2$.