Question

Question #786af

Answer 1

`1/4(4a^3 + 6a^3 +4a^2 +a +1)`

Explanation:

I'm interpreting the question as:
`lim_(h to 0)(h*((a+0h)^3+(a+1h)^3+cdots+(a+(n-1)h)^3))`

`lim_(h to 0)sum_(k=0)^(n-1)(h*f(a+k*h))` where `f(x)=x^3`.

This is a left Riemann sum for the function `f(x)=x^3` on the interval from `a` to `a+1`. In this case `h=(a+1-a)/n = 1/n`.

We can evaluate the limit by evaluating the equivalent definite integral:

`int_a^(a+1)x^3dx = [1/4x^4]_a^(a+1) = 1/4((a+1)^4-a^4)`

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

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