Math · Calculus · Worked example

Compare sums for x² on [0,1]

Use two rectangles and compare with the integral.

Choose the partition

Two equal pieces have width 1/2.

x_{0} = 0, x_{1} = \frac{1}{2}, x_{2} = 1

The increasing curve makes the left estimate low and the right estimate high.

L_{2} = \frac{1}{8}, R_{2} = \frac{5}{8}

Sample at 1/4 and 3/4.

M_{2} = \frac{1}{2} (\frac{1}{16} + \frac{9}{16}) = \frac{5}{16}

An antiderivative gives the exact comparison.

\int_{0}^{1} x^{2} d x = \frac{1}{3}

Left 0.125, right 0.625, midpoint 0.3125; the integral is 1/3.

Use two midpoint rectangles for f(x)=x on [0,2].

Show the answer and explanation

Width is 1 and midpoint heights are 0.5 and 1.5. For a linear function the midpoint rule is exact.

Change an input and predict the result before checking it.

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.