Use limit of a Riemann sum to evaluate 4xx2dx Solutionh ban

Use limit of a Riemann sum to evaluate: (4x-x2)dx,

Solution

h =(b-a)/n = 3/n
sum = limn-> 3/n [f(2) + f(2+3/n) + f(2+6/n) + f(2+9/n)........]
sum = limn-> 3/n [8 - 4 + 4(2+3/n) - (2+3/n)^2 + 4(2+6/n) - (2+6/n)^2 + 4(2+9/n) - (2+9/n)^2 + ......]
sum = limn-> 3/n [4 + 8 + 12/n - 4 - 12/n - 9/n^2 + 8 + 24/n - 4 - 24/n - 36/n^2 + 8 + 36/n - 4 - 36/n - 81/n^2 + ...........]
sum = limn-> 3/n [4 + 4 + 4 +...(n-1)terms -9/n^2[1+4+9+...(n-1)terms]
sum = limn-> 3/n [4(n-1) - 9/n^2{n(n-1)(2n-1)}/6 ]
sum = limn-> [12(n-1)/n - 27/n^3{n^3(1-1/n)(2-1/n)/6]
apply limits
sum = [12 - 9]
sum = 3