8. find a definite integral that is equal to $limlimits_{n\rightarrowinfty}sumlimits_{i = 0}^{n…

8. find a definite integral that is equal to $limlimits_{n\rightarrowinfty}sumlimits_{i = 0}^{n - 1}(2+\frac{i}{n})^{2}cdot(\frac{1}{n})$.

8. find a definite integral that is equal to $limlimits_{n\rightarrowinfty}sumlimits_{i = 0}^{n - 1}(2+\frac{i}{n})^{2}cdot(\frac{1}{n})$.

Answer

Explanation:

Step1: Recall the definition of definite integral

The definite integral (\int_{a}^{b}f(x)dx=\lim_{n\rightarrow\infty}\sum_{i = 0}^{n - 1}f(x_{i})\Delta x), where (\Delta x=\frac{b - a}{n}) and (x_{i}=a + i\Delta x).

Step2: Identify (\Delta x) and (x_{i})

Given (\Delta x=\frac{1}{n}), so (b - a=1). And (x_{i}=2+\frac{i}{n}), comparing with (x_{i}=a + i\Delta x), we can see that (a = 2), then (b=a+\Delta x\times n=2 + 1=3).

Step3: Identify (f(x))

Since (f(x_{i})=(2+\frac{i}{n})^{2}), and (x_{i}=2+\frac{i}{n}), then (f(x)=x^{2}).

Answer:

(\int_{2}^{3}x^{2}dx)