Question

For the function given below, find a formula for the Riemann sum obtained by dividing the interval (0, 3) into n equal subintervals and us right-hand endpoint for each Then take a limit of this sum as c_{k}; n -> ∞ to calculate the area under the curve over [0, 3] . f(x) = 2x ^ 2 Write a formula for a Riemann sum for the function f(x) = 2x ^ 2 over the interval [0, 3]

Answer 1

Splitting up [0, 3] into
`n` equally-spaced subintervals of length
`\Delta x=\frac{3-0}n = \frac3n` gives the partition

`\left[0, \frac3n\right] \cup \left[\frac3n, \frac6n\right] \cup \left[\frac6n, \frac9n\right] \cup \cdots \cup \left[\frac{3(n-1)}n, 3\right]`

where the right endpoint of the
`i`-th subinterval is given by the sequence

`r_i = \frac{3i}n`

for
`i\in\{1,2,3,\ldots,n\}`.

Then the definite integral is given by the infinite Riemann sum

`\displaystyle \int_0^3 2x^2 \, dx = \lim_(n\to\infty) \sum_(i=1)^n 2{r_i}^2 \Delta x \\\\ ~~~~~~~~ = \lim_(n\to\infty) \frac6n \sum_(i=1)^n \left(\frac{3i}n\right)^2 \\\\ ~~~~~~~~ = \lim_(n\to\infty) (54)/(n^3) \sum_(i=1)^n i^2 \\\\ ~~~~~~~~ = \lim_(n\to\infty) (54)/(n^3)\cdot\frac{n(n+1)(2n+1)}6 = \boxed{18}`