Compute the scalar line integral.

Question

Compute the scalar line integral for () = on , where is oriented counterclockwise and comprises : the ellipse $\ \frac{^}{ }+ \frac{^}{}= $ in the first quadrant, : the line segment from $\ (, )$ to $\ (, )$

Answer 1

The secret to all these types of questions is to parametrise your function $f(x)$ and path we need to integrate over. You say in a comment that you have done the second part so, for the first part the ellipse:
Note that we parametrise $\frac{x^2}4 + \frac{y^2}9 = 1$ in the first quadrant by $x=2\cos(\theta) , y = 3\sin(\theta)$ with $\theta \in [0, \frac{\pi}2] $ -convince yourself that this parametrises the desired part of the ellipse!! Then the question reduces to integrating $f(x) = xy = 6\cos(\theta)\sin(\theta)$:
$\int_0^{\frac{\pi}2} 6\cos(\theta)\sin(\theta) d\theta$

This integral is now done with a simple application of the sine double angle formula, hope that helps!!