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Section 5.1 Area and Estimating with Finite Sums
Example 5.1.1.
What is the area between the \(x\)-axis and the graph of \(f(x) =x^2+1\) between \(x=0\) and \(x=1\text{?}\)
- Upper sum
\(\displaystyle \frac{1}{4}f(\frac{1}{4}) +\frac{1}{4}f(\frac{2}{4}) +\frac{1}{4}f(\frac{3}{4}) +\frac{1}{4}f(\frac{4}{4})\)
- Lower sum
\(\displaystyle \frac{1}{4}f(\frac{0}{4}) +\frac{1}{4}f(\frac{1}{4}) +\frac{1}{4}f(\frac{2}{4}) +\frac{1}{4}f(\frac{3}{4})\)
- Midpoint rule
\(\displaystyle \frac{1}{4}f(\frac{1}{8}) +\frac{1}{4}f(\frac{3}{8}) +\frac{1}{4}f(\frac{5}{8}) +\frac{1}{4}f(\frac{7}{8})\)
- Common structure
\(\displaystyle \Delta x f(c_1) + \Delta x f(c_2) + \Delta x f(c_3) + \cdots + \Delta x f(c_n)\)
