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I’m in Pre-Calc and I just finished up a practice assessment for the area under a curve, but I got them all wrong. I take this course online and I did it the way I showed. Even my math teacher told me I was solving it correctly when I asked him. I’m going to post my solution to one of the problems. Could you tell me where I went wrong’ Thanks in advance.

Use limits to evalute each integral for questions 3-4.
3. 5’1 x^3 dx

Note:
1. The limit is for as x approaches infinity
2. “n” is on top of ‘ and “i = 1″ is on the bottom

(delta x) = (b-a)/n = (5-0)/n = 5/n
(xi) = a + i(delta x) = 0 + i(5/n) = 5i/n

5’0 x^3 dx
= lim ‘ ((xi)^3 (delta x)
= lim ‘ ((5i/n)^3 (5/n))
= lim ‘ (25i^3/n^4)
= lim [ ((25*1^3)/n^4) + ((25*2^3)/n^4) + ... + ((25*n^3)/n^4) ]
= lim 25/n^4 [ 1^3 + 2^3 + ... + n^3 ]
= lim (25/n^4) ((n^2(n+1)^2)/4)
= lim ((25n^2(n+1)^2)/4n^4)
= lim ((25(n+1)^2)/4n^2)
= lim ((25(n^2+2n+1))/4n^2)
= lim (25n^2+50n+1)/4n^2
= lim (25n^2/4n^2) + lim (50n/4n^2) + lim (1/4n^2)
= lim (25/4) + lim (25/2n) + lim (1/4n^2)
= 25/4 + 0 + 0
= 25/4

delta x = 1/n
xi = 1i/n

1’0 x^3 dx
= lim ‘ (xi)^3 (delta x)
= lim ‘ (1i/n)^3 (1/n)
= lim ‘ (1i^3/n^4)
= lim [ ((1*1^3)/n^4) + ((1*2^3)/n^4) + ... + ((1*n^3)/n^4) ]
= lim 1/n^4 [ 1^3 + 2^3 + ... + n^3 ]
= lim (1/n^4)((n^2(n+1)^2)/4)
= lim (n^2(n+1)^2)/4n^4)
= lim (n+1)^2/4n^2
= lim (n^2+2n+1)/4n^2
= lim (n^2/4n^2) + lim (2n/4n^2) + lim (1/4n^2)
= lim (1/4) + lim (1/2n) + lim (1/4n^2)
= 1/4 + 0 + 0
= 1/4

5’1 = 5’0 – 1’0 = 25/4 – 1/4 = 24/4 = 6

So my answer was 6 units^2, but the correct answer was 156 units^2. What did I do wrong that could have it thrown off that bad’ Any help would be appreciated.