Double Integration without Anti-Derivatives

[LCKurtz]

Well I, for one, would not be satisfied with what you have done, and you shouldn't be either, even though you found the correct answer. Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg. You don't need trig functions or to rotate anything. And it is also easy to calculate the area exactly and to write down an exact formula for your integral. And the answers just involve simple fractions. You would only get partial credit from me.

 

[SammyS]

Well I, for one, would not be satisfied with what you have done, and you shouldn't be either, even though you found the correct answer. Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg. You don't need trig functions or to rotate anything. And it is also easy to calculate the area exactly and to write down an exact formula for your integral. And the answers just involve simple fractions. You would only get partial credit from me.

Yes, LC makes an excellent point here.

According to the Original Post, the instructions for doing this problem included the following.

"Use deep conceptual understanding the insight (and no antiderviative calculations!) to reduce the iterated integral below to a simple algebraic expression depending on the parameters a, b, and c:"

(emphasis added by me)​

 

[CallMeShady]

Well I, for one, would not be satisfied with what you have done, and you shouldn't be either, even though you found the correct answer. Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg. You don't need trig functions or to rotate anything. And it is also easy to calculate the area exactly and to write down an exact formula for your integral. And the answers just involve simple fractions. You would only get partial credit from me.

I can calculate the area exactly without rotation; however, I could not think of a way to find the centroid of the triangle in the given orientation in the question. Hence, by making a simple rotation, it just made my life easier.

 

[LCKurtz]

Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg.

I can calculate the area exactly without rotation; however, I could not think of a way to find the centroid of the triangle in the given orientation in the question. Hence, by making a simple rotation, it just made my life easier.

Didn't I just tell you how to find it?