Please check if i am right - definite inegral

[DemiMike]

The question:
Use the definiton of the definite inegral (with right hand rule) to evaluate the following integral. Show work please
Can NOT use shortcut method.. must be the long process

Function:
1
S (3x^2 - 5x - 6) dx
-4

Work:
∫[-4,1] (3x^2 - 5x - 6) dx =
lim[n-->∞] 5/n ∑[i=1 to n] {3(-4 + 5/n)² - 5(-4 + 5/n) - 6} =
lim[n-->∞] 5*∑[i=1 to n] (48/n - 120i/n² + 75i²/n³ + 20/n -25i/n² - 6/n) =
lim[n-->∞] 5*∑[i=1 to n] (62/n - 145i/n² + 75i²/n³) =
lim[n-->∞] 5[62n/n - 145n(n+1)/(2n²) + 75n(n+1)(2n+1)/(6n³)] =
5(62 - 145/2 + 25) = 72.5

∑[i=1 to n] 1 = n
∑[i=1 to n] i = n(n+1)/2
∑[i=1 to n] i² = n(n+1)(2n+1)/6


Please check if this is correct and let me know

 

[Dick]

That looks ok to me. Even if you can't use the 'shortcut method' to solve the problem, you can always use the shortcut method to check your work.

 

[DemiMike]

it looks ok to me to =P, but i wanted to make sure if it's completely right

-thanks =)
which shortcut formula would i have to use

 

[Dick]

it looks ok to me to =P, but i wanted to make sure if it's completely right

-thanks =)
which shortcut formula would i have to use

I was assuming the 'shortcut' method was finding the antiderivative and evaluating it between the two limits. If you haven't learned that yet, then never mind.