use the limit process 2 find teh area of a region b/w teh graph of teh function and the x-axis over (read on!)

Question

the specified interval: f(x) = 16 - x^2, [2,4]

Answer 1

The region has a base of 4-2 that's divided into n pieces, so each piece has a width of 2/n. Each piece has a height that's f(2 + 2i/n) = 16 - (2 + 2i/n)² = 16 - 4 - 8i/n - 4i²/n² = 12 - 8i/n - 4i²/n².

So the area is lim (n→∞) of Σ (i=1 to n) of 2/n [12 - 8i/n - 4i²/n²]

for bevity, let's just write lim and Σ.

lim Σ [ 24/n - 16i/n² - 8i²/n^3] =
lim [24 - 16(n²/2 + n/2)/n² - 8(2n^3/6 +3n²/6 +n/6)/n^3] =
lim (n→∞) [24 - 8 - 8/n - 8/3 - 4/n - 4/3n²] =
16 - 0 - 8/3 - 0 - 0 =
40/3

Answer 2

Approach the limit from both the left and the right.