Question about the limits of integration in a change of variables

[Chenkel]

Hello everyone,

If I have an integral $\int_0^r \sqrt{(r^2 - x^2)}dx$ and I'm integrating across the first quadrant to get the area of the first quater of a circle.

And I change variables with $x = r\cos{\theta}$ and $dx = -{r}\sin{\theta}{d\theta}$

And I form a new integral that's $-\int_{\frac {\pi}{2}}^0 \sqrt{(r^2 - ({r\cos{\theta}})^2)}r\sin{\theta}{d\theta}$

Did I calculate the limits of integration correctly? Because when x is 0 the arc from the x axis to y axis is $\frac {\pi}{2}$ and when x is r the arc is 0.

I got positive ${\frac {\pi} 4} r^2$ for the first quadrant area (1 quarter of the circle using these equations) so I imagine my approach should be working but I wonder about the rigorourness of the change of limits of integration in my approach.

Let me know what you think!

Thank you!

 

[PeroK]

Yes that's fine. Note that, by definition, $\int_a^b = -\int_b^a$. You could have sorted out the limits that way. In general, using a decreasing function in your substitution is fine.

Note that you could have used sine instead of cosine. Which might have been slightly simpler.

 

[Chenkel]

Yes that's fine. Note that, by definition, $\int_a^b = -\int_b^a$. You could have sorted out the limits that way. In general, using a decreasing function in your substitution is fine.

Note that you could have used sine instead of cosine. Which might have been slightly simpler.

Thank you!