Homework Help

Use this definition to find an expression for the area under the graph of fas a limit....

user profile pic

frdsmith | Student, Grade 11 | eNoter

Posted April 29, 2012 at 5:05 AM via web

dislike 0 like

Use this definition to find an expression for the area under the graph of fas a limit. Do not evaluate the limit. f(x)=(lnx)/(x) 

5 ≤ x ≤ 12

1 Answer | Add Yours

user profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted April 29, 2012 at 4:44 PM (Answer #1)

dislike 1 like

You need to get the area under the given curve, hence, you need to evaluate the limit as the number of rectangles under the curve approaches to infinite.

You need to consider the width of rectangle as `Delta x`  and the height of rectangle as `f(x_i)` .

You need to evaluate in case of n rectangles such that:

`Delta x = (12-5)/n =gt Delta x = 7/n`

You need to evaluate the area of rectangle using the right points such that:

`A = f(x_i)*Delta x`

`x_i = 5 + i*Delta x =gt x_i = 5 + i*7/n`

Substituting `x_i ` in `f(x_i)`  yields:

`f(x_i) = (ln(5 + (7i)/n))/(5+(7i)/n)`

`A = f(x_i)Delta x = (ln(5 + (7i)/n))/(5+(7i)/n)*(7/n)`

Hence, evaluating the area under the curve as the number of rectangles approaches to infinite yields:

`A = lim_(n-gtoo) sum_(i=1)^n(7ln(5 + (7i)/n))/(5n+7i)`

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes