Line Integral Setup

Question

I have a line integral for homework, and I want to make sure I'm setting it up correctly before I begin solving. Here is the question...

Evaluate the line integral $\int_C xsin(y)ds$ where $C$ is the line segment from (0,1) to (3,5).

I have that $y=4/3x + 1$ with $x=t$ and $y = 4/3t + 1$ which makes

$ds=sqrt(1+(4/3)^2)dt = 5/3dt$

So finally, I have $\int_0^3tsin(4/3t+1)(5/3)dt$.

is this right?

Answer 1

First parametrize the path $C(t):=(3-0,5-1)\cdot t+(0,1)=(3t,4t+1)$ for $t \in [0,1]$. Now differentiate this expression $\dot{C}(t)=(3,4)$ and calculate the length of this vector $|\dot{C}(t)|=\sqrt{3^2+4^2}=5$, this is your $ds/dt$.

Now substitute everything into your integral:

$$\int_{0}^{1}(3t\sin(4t+1))\cdot 5dt$$