Question

Difficult Calculus question, please help me understand it. I am following along intently and I want to learn.

Answer 1

The definite integral expressed as Riemann sum is

`\int ^b_af(x)dx=\lim _(n\rightarrow\infty)\sum ^n_(i=1)f(x_i)\Delta x,\text{ where }\Delta x=(b-a)/(n),\text{ and }x_i=a+i\Delta x`

If we want to find the graph of the function f(x) from x = 1 up to x = 5, then its definite integral is

`\begin{gathered} \int ^b_af(x)dx \\ \\ \text{Substitute} \\ f(x)=x^2 \\ a=1,b=5 \\ \\ \text{THEN} \\ \int ^5_1(x^2)dx \end{gathered}`

Converting that definite integral into Riemann sum we have

`\begin{gathered} \int ^5_1(x^2)dx=\lim _(n\rightarrow\infty)\sum ^n_(i=1)(1+i(5-1)/(n))^2((5-1)/(n)) \\ =\lim _(n\rightarrow\infty)\sum ^n_(i=1)(1+i(4)/(n))^2(4)/(n) \\ =\lim _(n\rightarrow\infty)\sum ^n_(i=1)(1+(4i)/(n))^2(4)/(n) \end{gathered}`
`\begin{gathered} \text{Therefore, the area under the graph of the function }f(x)=x^2\text{ from }x=1\text{ to }x=5 \\ \text{is the Riemann Sum} \\ \lim _(n\rightarrow\infty)\sum ^n_(i=1)(1+(4i)/(n))^2(4)/(n) \end{gathered}`