Consider the integral given by $$\int_{1}^{\,2} {(2 x^3)}\, dx$$ Using the following definition of the integral $$\int_a^b f(x)dx=\lim_{n\rightarrow\infty}\sum_{i=1}^n f(x_i)\Delta x$$ where $x_i$ are right hand endpoints.

(a) Determine $\Delta x$ and $x_i.$

$\Delta x =$

$x_i =$

(b) Using the definition mentioned above, evaluate the integral.

Value of integral: