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Evaluate the integral, where R is the triangle with vertices at….?

Integral: R ∫(2x+y)^2 dAVertices: (-8,0) , (0,8) and (8,0)Give exact answer in form of a fraction.I know you have to do 2 different integrals for each side of the triangle but I am confused... Please help! Thank you!
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Dana Herzog

Feb 20, 2025

R is bounded by the lines y = 0, x+y = 8, and x-y = -8.==> x = y-8 to x = 8-y with y in [0, 8].(So, we only need to write one double integral in this manner.)So, we obtain∫(y = 0 to 8) ∫(x = y-8 to 8-y) (2x + y)^2 dx dy= ∫(y = 0 to 8) (1/6)(2x + y)^3 {for x = y-8 to 8-y} dy= ∫(y = 0 to 8) (1/6) [(2(8 - y) + y)^3 - (2(y - 8) + y)^3] dy= ∫(y = 0 to 8) (1/6) [(16 - y)^3 - (3y - 16)^3] dy = (-1/6) ∫(y = 0 to 8) [(y - 16)^3 + (3y - 16)^3] dy= (-1/6) [(1/4)(y - 16)^4 + (1/3)(1/4)(3y - 16)^4] {for y = 0 to 8}= (-1/72) [3(y - 16)^4 + (3y - 16)^4] {for y = 0 to 8}= 10240/3.Double check:http://www.wolframalpha.com/input/?i=%E2%88%AB%28y...I hope this helps!... Show More
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