Question about the limits of integration in a change of variables

  • #1
Chenkel
482
108
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!
 
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  • #2
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.
 
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  • #3
PeroK said:
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!
 

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