Question

Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval.
y = 16 − x^2, [−4, 4]

I completely got lost while doing this.

Answer 1

Was this relevant to a Cal I/II course in which you're expected to use the limit of Riemann Sums to find the area? Is that what you're referring to when you say the "limit process"?

Answer 2

\[\Large\lim_{n\to inf}~\sum_{i=1}^n~(16-(-4+i\frac{4+4}{n}))\frac{4+4}{n}\]

Answer 3

is the (4+4)/n because of (4-(-4))/n?

Answer 4

\[\Large\lim_{n\to inf}~\sum_{i=1}^n~(16-(-4+\frac{8i}{n})^2)\frac{8}{n}\]
\[\Large\lim_{n\to inf}~\sum_{i=1}^n~\frac{16(8)}{n}-\frac{8}{n}(-4+\frac{8i}{n})^2\]
\[\Large\lim_{n\to inf}~\sum_{i=1}^n~\frac{16(8)}{n}-\frac{8}{n}(16+\frac{64i^2}{n^2}-\frac{32i}{n})\]
\[\Large\lim_{n\to inf}~\sum_{i=1}^n~\frac{16(8)}{n}-\frac{16(8)}{n}-\frac{64(8)i^2}{n^3}+\frac{32(8)i}{n^2}\]

Answer 5

yes, the length of the interval partioned into n parts

Answer 6

those larger values can most likely be factored out as constants

Answer 7

\[\Large\lim_{n\to inf}~\sum_{i=1}^n~-\frac{2^9i^2}{n^3}+\frac{2^8i}{n^2}\]

Answer 8

ok, so at the second step with the (-4 +(8i)/n) that's (a+\[Deltax\i]?

Answer 9

correct

Answer 10

I'm sorry. I'm really slow at this stuff. XD

Answer 11

sok, it took some time looking at it and practicing before i got good enough to forget most of it lol

Answer 12

I'm still confused. the example video for the question prompted that I change the interval to [0,4] and then multiply the very end result by 2 to get the area. I did exactly like it did but it didn't work out.

Answer 13

you can do that if you wish

Answer 14

the end results i believe are: 256/3 either way

Answer 15

O.O I was so off. I got 320/3 the first time and then 128/3 the second time. O.O

Answer 16

ooooooh nvm. the second time, I forgot to multiply it by 2. XD

Answer 17

:) its always the algebra that gets ya in the end

Answer 18

thank you ^^ math is probably one of my weakest point (beside chemistry and biology)