Question

Approximate the area under the

Answer 1

Split up the interval [0, 3] into 3 equally spaced subintervals of length $\Delta x = \frac{3-0}3 = 1$. So we have the partition

[0, 1] U [1, 2] U [2, 3]

The left endpoint of the $i$-th subinterval is

$\ell_i = i - 1$

where $i\in\{1,2,3\}$.

Then the area is given by the definite integral and approximated by the left-hand Riemann sum

$\displaystyle \int_0^3 f(x) \, dx \approx \sum_{i=1}^3 f(\ell_i) \Delta x \\\\ ~~~~~~~~~~ = \sum_{i=1}^3 (i-1)^3 \\\\ ~~~~~~~~~~ = \sum_{i=0}^2 i^3 \\\\ ~~~~~~~~~~ = 0^3 + 1^3 + 2^3 = \boxed{9}$