Calculus: Find an expression for the area under the graph as a limit?

Question

Use the definition of a definite integral to find an expression for the area under the graph of the function ( f(x) = x^2 + \sqrt{1 + 2x} ) over the interval ( [5, 7] ). Please do not evaluate the limit. I found a helpful resource that gives a general idea of what the answer should look like: http://www.webassign.net/scalcet7/5-1-definition-0. However, I've tried several different approaches, and it seems that WebAssign is being quite specific about the format. Any guidance would be greatly appreciated!

Anonymous · Accepted Answer

Using right endpoints with n rectangles of equal width along [5, 6]:

Δx = (7 - 5)/n = 2/n.

So, x(i) = a + iΔx = 5 + 2i/n [for right endpoint approx.]

Hence, the area (via right endpoints) equals

lim(n→∞) Σ(i = 1 to n) f(a + iΔx) Δx

= lim(n→∞) Σ(i = 1 to n) f(5 + 2i/n) * (2/n)

= lim(n→∞) Σ(i = 1 to n) [(5 + 2i/n)^2 + √(1 + 2(5 + 2i/n))] * (2/n).

I hope this helps!

Mr. Enos Barton PhD · Answer

Just do the anti-derivative to short cut this 
& also an exact answer.
int_5^7 [x^2 + sqrt(1 + 2x)] dx
[(x^3 / 3)] + [(2x + 1)^(3/2) / 3]
F(7) - F(5)
[(7^3 / 3)] + [(27 + 1)^(3/2) / 3] - {[(5^3 / 3)] + [(25 + 1)^(3/2) / 3]}
[(343 / 3)] + [(15)^(3/2) / 3] - [(125/ 3)] - [(11)^(3/2) / 3]
(218/3) + {[15sqrt(15) - 11sqrt(11)]/3}
{(218) + [15*sqrt(15)] - [11*sqrt(11)]}/3

Nikki Wolff · Answer

79.8706

General Collier MD

Calculus: Find an expression for the area under the graph as a limit?

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