Calculus

posted by .

Evaluate the definite integral from [0,4] 4x^2 dx, by using its definition as a limit of approximating sums.

First, I solve analytically so I know the answer I am trying to reach: 4/3 * x^3 over [0,4] = 4/3 * 4^3 = 256/3

Now, by approximating sums, I can get:

256 * lim(n -> infinity) of sum(j=0 to n-1) of j^2/n^3

I can use the computer to solve this and I reach the correct answer. In Mathematica Alpha notation: 256 * lim sum j^2/n^3, j=0..n as n -> infinity

How do I solve that limit by hand? Or is there a better way to solve the original problem?

  • Calculus -

    You need to derive the formula for:

    Sum from j = 0 to N of j^2

    There are many different ways to do this, the more advanced math you know, the simpler it gets :).

    An elementary method is to consider summing (j+1)^3 - j^3 instead of j^2. Obviously, if you sum a function of the form f(j+1) - f(j), all the terms except the first and last one will cancel:

    Sum from j = 0 to N of [f(j+1) - f(j)] =

    f(N+1) - f(0)

    If we choose f(j) = j^3, then:

    f(j+1) - f(j) =

    (j+1)^3 - j^3 =

    3 j^2 + 3 j + 1

    So, if you know the summation of j from zero to N, you can find the summation of j^2. Of course, you can find the formula for the summation of j in the same way by taking f(j) = j^2.

  • Calculus -

    You are awesome Iblis. That limit was the tricky part. I followed your proof and it makes perfect sense. Thanks

  • Calculus -

    I meant that summation was the tricky part (not limit). thanks again.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calculus

    Our professor wants us to evaluate the limits analytically without using a table or a graph, and if it doesn't exist we must describe the behavor near the limit point. I'm not sure how to evaluate each side of a limit separately without …
  2. Calculus Fundamental Theorem

    Evaluate the definite integral. function: x+13 with respect to variable x lower limit:0 upper limit:22
  3. Calculus Fundamental Theorem

    Evaluate the definite integral. function: (t+8)(t^2+3) with respect to variable t lower limit: -sqrt(2) upper limit: sqrt(2)
  4. Calculus Area between curves

    Evaluate the definite integral: sqrt(8-2x) lower limit=-7 upper limit=0 I got -(1/3)(8-2x)^(3/2) and it was wrong. Please Help! Thanks in advance!
  5. Calculus Help Please Urgent!!!

    Prove that the integral on the interval [a,b] of x is equal (b^2-a^2)/2 integral a to be (x)dx = (b^2-a^2)/2 using the definition of a Definite Integral. This is the limit of a sum approach. show steps please!!! Thank you!!!
  6. math

    express the limit as a definite integral. Do Not Evaluate the integral. lim n->infinity (1+i(3/n))^2(3/n) explain the answer please
  7. Calculus (urgent help)

    consider the function f that is continuous on the interval [-5,5] and for which the definite integral 0(bottom of integral sign) to 5(top of integral sign) of f(x)dx=4. Use the properties of the definite integral to evaluate each integral: …
  8. calculus

    consider the function f that is continuous on the interval [-5,5] and for which the definite integral 0(bottom of integral sign) to 5(top of integral sign) of f(x)dx=4. Use the properties of the definite integral to evaluate each integral: …
  9. calculus (please with steps and explanations)

    consider the function f that is continuous on the interval [-5,5] and for which the definite integral 0(bottom of integral sign) to 5(top of integral sign) of f(x)dx=4. Use the properties of the definite integral to evaluate each integral: …
  10. calculus

    Evaluate lim (1³ +2³ +3³ +…+ n3)/n^4 n →∞ by showing that the limit is a particular definite integral and evaluating that definite integral.

More Similar Questions