Question

(a) Determine Δx and xi. Δx= xi= (b) Using the definition mentioned above, evaluate the integral. Value of integral: ?

Answer 1

Explanation:

`Delta x = (b-a)/n = (2-1)/n = 1/n`

`x_i = a+iDeltax` for `i = 0, 1,2,3 ,. . . n`

`x_i = 1+i/n`

`f(x_i) = 2(1+i/n)^3 = 2(1+3i/n+3i^2/n^2+i^3/n^3)`

`sum_(i=1)^n f(x_i) Delta x`

`= sum_(i=1)^n2(1+i/n)^3 (1/n)`

`= 2sum_(i=1)^n(1+i/n)^3 (1/n)`

`= 2sum_(i=1)^n (1+3i/n+3i^2/n^2+i^3/n^3)(1/n)`

`= 2[sum_(i=1)^n (1/n+3i/n^2 +3i^2/n^3+i^3/n^4)]`

`= 2sum_(i=1)^n 1/n +2sum_(i=1)^n3i/n^2 +2sum_(i=1)^n3i^2/n^3+2sum_(i=1)^ni^3/n^4`

`= 2/nsum_(i=1)^n 1 +6/n^2 sum_(i=1)^n i +6/n^3 sum_(i=1)^n i^2+2/n^4 sum_(i=1)^n i^3`

`= 2/n(n) + 6/n^2 ((n(n+1))/2) +6/n^3 ((n(n+1)(2n+1))/6) + 2/n^4 ((n^2(n+1)^2)/4)`

`= 2+3((n(n+1))/n^2) + ((n(n+1)(2n+1))/n^3)+ 1/2 ((n^2(n+1)^2)/n^4)`

So

`int_0^1 2x^3 dx = lim_(nrarroo) sum_(i=1)^n f(x_i) Delta x`

`= lim_(nrarroo) ( 2+3((n(n+1))/n^2) + ((n(n+1)(2n+1))/n^3)+ 1/2 ((n^2(n+1)^2)/n^4)`

`= 2+(3)+(2)+1/2(1) = 7 1/2`