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

Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.

f(x) = x^2 + sqrt(1 + 2x), 5 ≤ x ≤ 7

http://www.webassign.net/scalcet7/5-1-definition-0...

The link should give a general idea of what the answer should look like. I've tried a few different answers, but I think webassign is being picky =(

Sorry, I shoul'dve pointed it out, but the answer needs to be in "lim--> sigma ....." form. So I need the step before solving for the actual answer.

3 Answers

  • 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!

  • 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

  • 79.8706

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