Integrate along curve, book has wrong answer?

[Addez123]

Homework Statement

$$\int _C y ds$$
where C is determined by
$$x^2+y^2=a^2, y >= 0$$

Relevant Equations

Math

So it's basically a half circle with radius a.
y = asin(t)
$$\int_0^{\pi} asin(t) dt = -acos(t) |_0^{\pi} = 2a$$

The book says the answer is $2a^2$, but maybe that's wrong?

 

[Mark44]

Homework Statement:: $$\int _C y ds$$
where C is determined by
$$x^2+y^2=a^2, y >= 0$$

So it's basically a half circle with radius a.
y = asin(t)
$\int_0^{\pi} asin(t) dt = -acos(t) |_0^{\pi} = 2a$
The book says the answer is $2a^2$, but maybe that's wrong?

I believe the book's answer. Notice that the integral includes ds, not dt.
Note that $ds = \sqrt{(dx/dt)^2 + (dy/dt)^2}dt$

 

[Addez123]

Thanks! Calculating dS was what I was missing, now it checks out!