Find area using limits

find the area from [0,3] of 4-(1/4)x^2 using the limit definition.
Please show work.

Start by dividing [0, 3] into n equal segments with width Δx = (3 - 0)/n = 3/n.

Using right endpoints, the area of the n rectangles equals
Σ(k = 1 to n) f(0 + k Δx) Δx
= Σ(k = 1 to n) f(3k/n) * (3/n)
= Σ(k = 1 to n) [4 - (1/4)(3k/n)^2] * (3/n), since f(x) = 4 - (1/4)x^2
= (3/(4n)) Σ(k = 1 to n) (16 - 9k^2/n^2)
= (3/(4n)) [16 Σ(k = 1 to n) 1 - (9/n^2) Σ(k = 1 to n) k^2]
= (3/(4n)) [16n - (9/n^2) * n(n+1)(2n+1)/6], via summation formulas
= (3/4) [16 - (3/2) * (n+1)/n * (2n+1)/n]
= (3/4) [16 - (3/2) * (1 + 1/n) * (2 + 1/n)].

The desired area of the region is now obtained by letting n → ∞:
A = (3/4) [16 - (3/2) * 1 * 2] = 39/4.
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Double check:
A = ∫(x = 0 to 3) (4 - x^2/4) dx = (4x - x^3/12) {for x = 0 to 3} = 39/4.

I hope this helps!