Infinite Riemann sum of $x^3$

Question

Write

$$\int_{1}^{3} x^3 dx$$ as a riemann sum.

Here is what I thought:

$$\Delta(x) = \frac{2}{n}$$

$$f(x) = (\Delta(x)k)^3 = \left(\frac{2k}{n}\right)^3$$

$$I = \int_{1}^{3} x^3dx = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{2}{n}\cdot \left(\frac{2k}{n}\right)^3$$

But the sum evaluates to $4$ instead of $20$ which is the actual value...?

Answer 1

You forgot 1:

$$I = \int_{1}^{3} x^3dx = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{2}{n}\cdot \left(1+\frac{2k}{n}\right)^3$$