Question

Use the definition to find an expression for the area under the curve y = x3 from 0 to 1 as a limit. lim n→∞ n i = 1 R (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13 + 23 + 33 + + n3 = n(n + 1) 2 2

Answer 1

The summand (R?) is missing, but we can always come up with another one.

Divide the interval [0, 1] into $n$ subintervals of equal length $\dfrac{1-0}n=\dfrac1n$:

$[0,1]=\left[0,\dfrac1n\right]\cup\left[\dfrac1n,\dfrac2n\right]\cup\cdots\cup\left[1-\dfrac1n,1\right]$

Let's consider a left-endpoint sum, so that we take values of $f(\ell_i)={\ell_i}^3$ where $\ell_i$ is given by the sequence

$\ell_i=\dfrac{i-1}n$

with $1\le i\le n$. Then the definite integral is equal to the Riemann sum

$\displaystyle\int_0^1x^3\,\mathrm dx=\lim_{n\to\infty}\sum_{i=1}^n\left(\frac{i-1}n\right)^3\frac{1-0}n$

$=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=1}^n(i-1)^3$

$=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=0}^{n-1}i^3$

$=\displaystyle\lim_{n\to\infty}\frac{n^2(n-1)^2}{4n^4}=\boxed{\frac14}$