Question

Express the integral as a limit of Riemann sums. Do not evaluate the limit. (Use the right endpoints of each subinterval as your sample points.)

int 1 between 5 x/2+3^3*dx

Answer 1

`\mathbf{4 \lim \limits _(n \to \infty) \sum \limits ^n_(i=1) \Big ( (n(n+4i))/(2n^3 +(n+4i)^3) \Big )}`

Explanation:

Given integral:

`\int ^5_1 (x)/(2+x^3) \ dx`

`\mathbf{Using \ Riemann \ sums; \ we \ have: }`

`\int ^b_a \ f(x) \ dx = \lim_(n \to \infty) \sum \limits ^n_(i =1) \ f( a + i \Delta x) \Delta x`

`here; \ \Delta x = (b-a)/(n)`

∴

`\int ^5_1 (x)/(2+x^3) \ dx = f(x) = (x)/(2+x^3)`

`\implies \Delta x = (5-1)/(n) =(4)/(n)`

`f(a + i \Delta x ) = f ( 1 + (4i)/(n))`

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

`\lim_(n \to \infty) \sum \limits ^n_(i=1) \ f(a + i \Delta x) \Delta x = \lim_(n \to \infty) \sum \limits ^n_(i=1) \Big ( (n^2(n+4i))/(2n^3 +(n+4i)^3) \Big )(4)/(n)`

`\mathbf{= 4 \lim \limits _(n \to \infty) \sum \limits ^n_(i=1) \Big ( (n(n+4i))/(2n^3 +(n+4i)^3) \Big )}`