Express the integral below as the limit of a Riemann Sum. Be sure to specify your choice of ci's, and deltax 5 x^47x 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^47x dx

0
DeltaX=
ci=
lim n
n>infinity E
i=1
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1 Answer
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 = (50)/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_(ngtoo) sum_(i=1)^n(3125i^4/(n^5)  175i/(n^2))`