Question

Approximate the area under the

function between a and b using a
left-hand sum with the given
number of intervals.
f(x) = x³
a=0
b=3
3 Intervals

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}`