Before we can compute the exact area under a curve, we need a way to approximate it. Riemann sums are the foundation: we divide the area into rectangles, sum their areas, and let the rectangles get narrower and narrower to approach the true area.
🎯 In this section you will learn
- The definition of a Riemann sum
- Left, right, and midpoint Riemann sums
- The trapezoidal rule for better approximation
- How increasing the number of rectangles improves accuracy
📌 The Basic Idea
To approximate the area under $y = f(x)$ from $x = a$ to $x = b$:
$$ \text{Area} \approx \sum_{i=1}^{n} f(x_i^*) \Delta x $$
where $\Delta x = \frac{b-a}{n}$ is the width of each rectangle, and $x_i^*$ is a sample point in the $i$-th subinterval.
💡 As $n \to \infty$, the approximation becomes exact
$$ \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x = \int_a^b f(x) \, dx $$
📊 Visualizing Riemann Sums
Both graphs show $f(x) = x^2$ on $[0, 4]$ with $n = 4$ rectangles. Notice how the left sum misses area above each rectangle, while the right sum overshoots.
Left Riemann Sum — n = 4
Approximation: 14.00
Exact: 21.33
Underestimate
Right Riemann Sum — n = 4
Approximation: 30.00
Exact: 21.33
Overestimate
📋 Types of Riemann Sums
| Type | Sample Point $x_i^*$ | Accuracy |
|---|
| Left Riemann Sum | Left endpoint of each subinterval | Underestimates when $f$ is increasing, overestimates when decreasing |
| Right Riemann Sum | Right endpoint of each subinterval | Overestimates when $f$ is increasing, underestimates when decreasing |
| Midpoint Riemann Sum | Midpoint of each subinterval | More accurate than left/right (error is smaller) |
| Trapezoidal Rule | Average of left and right sums | More accurate; approximates area using trapezoids instead of rectangles |
Approximate $\int_0^4 x^2 dx$ using a left Riemann sum with $n=4$ subintervals.
①
$\Delta x = \frac{4-0}{4} = 1$
Subintervals: $[0,1], [1,2], [2,3], [3,4]$
②
Left endpoints: $0, 1, 2, 3$
$f(0)=0$, $f(1)=1$, $f(2)=4$, $f(3)=9$
③
Sum
$\text{Area} \approx 1(0+1+4+9) = 14$
④
Compare with exact
Exact $\int_0^4 x^2 dx = \frac{64}{3} \approx 21.33$
Approximate $\int_0^4 x^2 dx$ using a right Riemann sum with $n=4$.
- $\Delta x = 1$, right endpoints: $1, 2, 3, 4$
- $f(1)=1$, $f(2)=4$, $f(3)=9$, $f(4)=16$
- $\text{Area} \approx 1(1+4+9+16) = 30$
- Exact = $21.33$ — right sum overestimates (since $f$ is increasing)
Approximate $\int_0^4 x^2 dx$ using a midpoint sum with $n=4$.
- $\Delta x = 1$, midpoints: $0.5, 1.5, 2.5, 3.5$
- $f(0.5)=0.25$, $f(1.5)=2.25$, $f(2.5)=6.25$, $f(3.5)=12.25$
- $\text{Area} \approx 1(0.25+2.25+6.25+12.25) = 21$
- Exact = $21.33$ — midpoint sum is very close!
Approximate $\int_0^4 x^2 dx$ using the trapezoidal rule with $n=4$.
- Average of left and right sums: $\frac{14 + 30}{2} = 22$
- Exact = $21.33$ — the trapezoidal rule is more accurate than left or right alone
⚠️ Important Notes
- For increasing functions: left sum underestimates, right sum overestimates.
- For decreasing functions: left sum overestimates, right sum underestimates.
- The midpoint sum is usually the most accurate among the three rectangle methods.
- As the number of rectangles $n$ increases, all Riemann sums approach the exact area.
🔍 Key Takeaways
- Riemann sums approximate area using rectangles: $\sum f(x_i^*) \Delta x$.
- Left sum uses left endpoints, right sum uses right endpoints, midpoint sum uses midpoints.
- The trapezoidal rule averages left and right sums (more accurate).
- As $n \to \infty$, the Riemann sum approaches the exact definite integral.
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