Can you help me rewrite this integral to cylindrical coordinates?

[deheerbeer]

I managed to solve the following problem:


by using the following identity (along with u-substitution and computation):

Now I'm wondering if it is possible to rewrite this integral in cylindrical, bypassing the need to memorise this integral. I can post my notes of this exercise, but they don't relate to my question directly.

I've tried to use the identity y = r sin θ, but I can't figure out how to rewrite the integral to cylindrical. Please let me know if more information would help you answer my question.

 

[Deleted member 4993]

I managed to solve the following problem:
View attachment 26979
View attachment 26980
by using the following identity (along with u-substitution and computation):
View attachment 26981
Now I'm wondering if it is possible to rewrite this integral in cylindrical, bypassing the need to memorise this integral. I can post my notes of this exercise, but they don't relate to my question directly.

I've tried to use the identity y = r sin θ, but I can't figure out how to rewrite the integral to cylindrical. Please let me know if more information would help you answer my question.

It will be very difficult to describe the boundary - a triangle - in polar coordinate!!

 

[HallsofIvy]

The region of integration is the triangle with vertices (0, 0), (a, 0), and (a, a).
In polar coordinates the point (a, 0) is on the line with angle \(\displaystyle \theta= 0\) and (a, a) is on the line with angle \(\displaystyle \theta= \pi/4\).

For any angle, \(\displaystyle \theta\), between those bounds, the ray extends from the origin to the vertical line \(\displaystyle y= r sin(\theta)= a\) so \(\displaystyle r= \frac{a}{sin(\theta)}\).

The integral, in polar coordinates, is \(\displaystyle \int_{\theta= 0}^{\pi/4} \int_{r= 0}^{a/sin(\theta)} \sqrt{a^2- r^2sin^2(\theta)} r dr d\theta\).

(Strictly speaking, this is in "polar coordinates", NOT "cylindrical coordinates" because it is a two dimensional problem, not three dimensional.)

 

[deheerbeer]

Thank you for your replies. They have confirmed my suspicions. I wish there was a way to more easily memorise such integrals as the one I mentioned.

 

[deheerbeer]

It will be very difficult to describe the boundary - a triangle - in polar coordinate!!

Is there another way to do this exercise, other than knowing this integral?



I cannot have a cheat list so I will need to memorise a lot of these for the exam, if that is the case. Luckily arcsin values are the same as sin