AP Calculus AB / Unit 6: Integration and Accumulation of Change
6.2 Approximating Areas with Riemann Sums
Notes
Approximate area with rectangles or trapezoids:
- Left sum: heights from left endpoints. Right sum: right endpoints. Midpoint sum: middles.
- Trapezoidal sum: average of left and right on each piece.
- For increasing $f$, left sums underestimate and right sums overestimate.
Practice
$f(0)=1$, $f(1)=3$, $f(2)=4$. Left Riemann sum on $[0,2]$ with 2 equal pieces:
$1(1)+1(3)=4$.
Using the same values, the trapezoidal sum is:
$\displaystyle \frac{1}{2}(1+3)+\frac{1}{2}(3+4)=2+3.5$.
For a decreasing function, a right Riemann sum:
Right endpoints are the lowest heights on each piece.
Ready for more? Take the Unit 6 test.