- #1
QuarkCharmer
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Homework Statement
a.) Use definition 2 to find an expression for the area under the curve y=x^3 from 0 to 1 as a limit.
b.)Evaluate the (above) limit using the sum of the cubes of the n integers.
Homework Equations
(\frac{n(n+1)}{2})^{2}
The Attempt at a Solution
For part a.) I wrote my limit like this:
\lim_{n \to \infty} \Sigma_{i=1}^{n}(\frac{i}{n})^{3}\frac{1}{n}The "Definition 2" they have listed just says:
A=\lim_{n \to \infty} R_{n} = \lim_{n \to \infty}(f(x_{1})\Delta x + f(x_{2})\Delta x + . . . + f(x_{n})\Delta x)
Now for part b, I understand that the formula is the sum of all cubes and so on. So I am thinking that the limit should look like this?
\lim_{n \to \infty}(\frac{n(n+1)}{2})^{2}
That should handle the limit and the sum of cubes, now I need each one to multiply by delta x right? So that it comes out to:
\lim_{n \to \infty}(\frac{n(n+1)}{2})^{2}\frac{1}{n}
..because the integral is from 0 to 1, so \Delta x = \frac{1-0}{n}
But I am not sure how to write it in this manner and take the limit from here yet. Is this correct so far? I should just simplify the expression after the limit and then take the limit?
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