# Express the integral  below as the limit of a Riemann Sum. Be sure to specify your choice of ci's, and deltax 5|     x^4-7x dx  | 0DeltaX= ci= lim              ...

Express the integral  below as the limit of a Riemann Sum. Be sure to specify your choice of ci's, and deltax
5
|     x^4-7x dx
|
0

DeltaX=

ci=

lim               n
n->infinity    E

i=1

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You should find the area under the given curve, hence, you need to evaluate the limit of sum of areas of n rectangles 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 = (5-0)/n =gtDelta x = 5/n`

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

`A = f(x_i)*Delta x`

`x_i = 0+i*Delta x =gtx_i =5i/n`

`f(x_i) = x_i^4 - 7x_i =gt f(x_i) = 625(i/n)^4 - 35i/n`

`A = (625(i/n)^4 - 35i/n)*(5/n)`

`A = 3125i^4/(n^5) - 175i/(n^2)`

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

`A = lim_(n-gtoo) sum_(i=1)^n(3125i^4/(n^5) - 175i/(n^2))`