Line Integrals - Part I — Question 5

Question 5

The piecewise smooth curve $C=C_1\cup C_2$ travels from $$A=(0,0)\quad\text{to}\quad B=(1,0)\quad\text{to}\quad D=(1,2)$$ along straight segments. Evaluate $$\int_C(xy+1)\,ds.$$

Tasks

1. Parametrize each segment.

2. Evaluate the integral on each piece and add the results.

3. Check the contribution using the integrand along each segment.

Question 5 – Solution

Strategy. A scalar line integral over a piecewise curve is the sum of the integrals over its smooth pieces.

Step 1: First segment

For $C_1$, use $\mathbf r_1(t)=\langle t,0\rangle$, $0\le t\le 1$. Then $ds=dt$ and $xy+1=1$, so $$\int_{C_1}(xy+1)\,ds=\int_0^1 1\,dt=1.$$

Step 2: Second segment For $C_2$, use $\mathbf r_2(u)=\langle 1,u\rangle$, $0\le u\le 2$. Then $ds=du$ and $xy+1=u+1$, giving $$\int_{C_2}(xy+1)\,ds =\int_0^2(u+1)\,du =\left[\frac{u^2}{2}+u\right]_0^2=4.$$ Hence $$\boxed{\int_C(xy+1)\,ds=1+4=5}.$$

Verification On $C_1$ the constant value $1$ times length $1$ gives $1$. On $C_2$ the integrand rises linearly from $1$ to $3$, so its average $2$ times length $2$ gives $4$.