Answer:
To find the expression of the area under the graph f as limit,
Given that,
[tex]f(x)=x+\ln (x),4\leq x\leq9[/tex]
we get,
Consider the small rectangles in the area under the graph f, such that length is f(x) at x point and width is
[tex]\Delta x=\frac{b-a}{n}[/tex]
Where a and b are the end points and the n represents the number of partitions.(for n tends to infinity we can get the accurate area under the graph f)
The area of the graph under f is given by,
[tex]\lim _{n\to\infty}\sum ^n_{i\mathop=1}f(x_i)\Delta x[/tex][tex]=\lim _{n\to\infty}\sum ^n_{i\mathop{=}1}(x_i+\ln (x_i))\frac{9-4}{n}[/tex][tex]=\lim _{n\to\infty}\sum ^n_{i\mathop{=}1}(x_i+\ln (x_i))\frac{5}{n}[/tex][tex]=\lim _{n\to\infty}\sum ^n_{i\mathop{=}1}\frac{5}{n}(x_i+\ln (x_i))[/tex]
Answer is:
[tex]A=\lim _{n\to\infty}\sum ^n_{i\mathop{=}1}\frac{5}{n}(x_i+\ln (x_i))[/tex]
