Consider the function f(x) = x^2/3 - 7|. In this problem you will calculate inte
ID: 2876348 • Letter: C
Question
Consider the function f(x) = x^2/3 - 7|. In this problem you will calculate integral _0^4 (x^2/3 - 7) dx by using the definition integral _a^b f(x) dx = lim_ n rightarrow infinity [Sigma _i = 1^n f(x_i) Delta x]| The summation inside the brackets is which is the R_n| which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval. Calculate R_n| for f(x) = x^2/3 - 7| on the interval [0, 4]|and write your answer as a function of n| without any summation signs. Rn=| ___________ lim _n rightarrow infinity R_n =| ___________Explanation / Answer
[0 to 4]((1/3)x2 -7) dx
a =0,b=4
x=(b-a)/n =(4-0)/n =4/n
xi=a+ix =0+i(4/n) =(4i/n)
Rn=[n=1 to ]f(xi)x
Rn=[n=1 to ](((1/3)(4i/n)2)-7)(4/n)
Rn=[n=1 to ](((16/3)(i2/n2)) -7)(4/n)
Rn=[n=1 to ]((64/3)(i2/n3)) -(28/n)
Rn=((64/3)(n(n+1)(2n+1)/(6n3))) -(28n/n)
Rn=((64/18)(1(1+(1/n))(2+(1/n)))) -(28)
limn->Rn
limn->((64/18)(1(1+(1/n))(2+(1/n)))) -(28)
=((64/18)(1(1+0)(2+(0)))) -(28)
=((64/18)*2) -28
=(64/9) -28
=-188/9