Riemann sums estimate the area under a curve by replacing a smooth region with rectangles. This idea matters because area under a graph often represents a total amount, such as distance from velocity, accumulated charge from current, or total change from a rate. Left, right, and midpoint sums use different points in each subinterval to choose the rectangle heights.
Comparing them helps students understand both approximation and the meaning of a definite integral.
To build a Riemann sum on [a,b], divide the interval into n equal pieces of width Δx = (b - a)/n. A left sum uses the function value at the left edge of each piece, a right sum uses the right edge, and a midpoint sum uses the center. For an increasing function, left sums underestimate the area and right sums overestimate it, while midpoint sums are often more balanced.
As n increases, the rectangles get thinner, the gaps and overlaps shrink, and the sum approaches the exact integral.
Key Facts
- Subinterval width: Δx = (b - a)/n
- Left Riemann sum: L_n = Δx[f(x_0) + f(x_1) + ... + f(x_{n-1})]
- Right Riemann sum: R_n = Δx[f(x_1) + f(x_2) + ... + f(x_n)]
- Midpoint Riemann sum: M_n = Δx[f((x_0 + x_1)/2) + ... + f((x_{n-1} + x_n)/2)]
- For an increasing function, L_n is an underestimate and R_n is an overestimate.
- The exact area is the limit of Riemann sums: ∫_a^b f(x) dx = lim n→∞ Σ f(x_i*)Δx.
Vocabulary
- Riemann sum
- A Riemann sum is an approximation of area under a curve using rectangles over smaller subintervals.
- Subinterval
- A subinterval is one smaller section of the full interval [a,b] after it has been divided into pieces.
- Delta x
- Delta x, written Δx, is the width of each rectangle when an interval is divided into equal parts.
- Sample point
- A sample point is the x-value chosen inside a subinterval to determine the height of a rectangle.
- Definite integral
- A definite integral gives the exact signed area under a function over a specified interval.
Common Mistakes to Avoid
- Using the wrong rectangle height for a left or right sum. A left sum uses the left endpoint of each subinterval, while a right sum uses the right endpoint.
- Forgetting to multiply by Δx. Adding only the function values gives a sum of heights, not an area estimate.
- Assuming left sums always underestimate and right sums always overestimate. That is true for increasing functions, but the pattern reverses for decreasing functions.
- Thinking more rectangles automatically means the exact answer has been reached. More rectangles usually improve the estimate, but the exact integral is reached only in the limit as n goes to infinity.
Practice Questions
- 1 For f(x) = x^2 on [0,4] with n = 4 equal subintervals, compute the left Riemann sum and the right Riemann sum.
- 2 For f(x) = 2x + 1 on [1,5] with n = 4 equal subintervals, compute the midpoint Riemann sum.
- 3 A function is decreasing and positive on [a,b]. Explain whether the left Riemann sum is an overestimate or an underestimate, and justify your answer using rectangle heights.