Calculus / Unit 7: Antiderivatives and the Integral
7.4 Riemann Sums
Notes
A Riemann sum estimates area under a curve by adding up rectangle areas: width $\displaystyle \Delta x = \frac{b - a}{n}$ times heights.
- Left sum uses the height at the left end of each subinterval, right sum the right end, midpoint sum the middle.
- For an increasing function, the left sum underestimates and the right sum overestimates.
- More rectangles give a better estimate.
- Example: $f(x) = x^2$ on $[0, 4]$ with $n = 4$ (width 1). Left sum $= 0 + 1 + 4 + 9 = 14$.
Practice
Find the left Riemann sum for $f(x) = x^2$ on $[0, 4]$ with 4 equal subintervals.
Width 1, heights at $x = 0, 1, 2, 3$: $0 + 1 + 4 + 9 = 14$.
Find the right Riemann sum for $f(x) = x^2$ on $[0, 4]$ with 4 equal subintervals.
Heights at $x = 1, 2, 3, 4$: $1 + 4 + 9 + 16 = 30$.
Find the midpoint sum for $f(x) = x^2$ on $[0, 4]$ with 2 equal subintervals.
Width 2, midpoints 1 and 3: $2(1 + 9) = 20$.
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