M171 Calculus I · Unit 4: Optimization & the integral · on Exam 4
§4.2 Riemann sums
Study notes for Riemann sums in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Divide $[a,b]$ into $n$ pieces of width $\Delta x = \frac{b-a}{n}$ with endpoints $x_i = a + i\Delta x$. A Riemann sum adds up rectangle areas $f(x_i^*)\Delta x$ using a chosen sample point in each piece.
- Left sum $L_n$ uses left endpoints, right sum $R_n$ uses right endpoints, midpoint sum $M_n$ uses midpoints. $M_n$ is usually most accurate.
- If $f$ is increasing, $L_n$ underestimates and $R_n$ overestimates the area; if decreasing, the reverse.
- Sigma notation: $\displaystyle\sum_{i=1}^{n} f(x_i)\Delta x$. As $n\to\infty$ all Riemann sums approach the definite integral.
Key formulas
Width
$$\Delta x = \frac{b-a}{n},\qquad x_i = a + i\,\Delta x$$
Left / right / midpoint
$$L_n = \sum_{i=0}^{n-1} f(x_i)\Delta x,\quad R_n = \sum_{i=1}^{n} f(x_i)\Delta x,\quad M_n = \sum_{i=1}^{n} f\!\left(\tfrac{x_{i-1}+x_i}{2}\right)\Delta x$$
Worked example
Estimate $\int_0^2 x^2\,dx$ with $L_4$ and $R_4$.
Show the solution
$\Delta x = 0.5$. $L_4 = 0.5\,(0 + 0.25 + 1 + 2.25) = 1.75$. $R_4 = 0.5\,(0.25 + 1 + 2.25 + 4) = 3.75$. The true value $\frac83\approx2.667$ lies between them because $x^2$ is increasing.
Common mistakes
- Using $n+1$ rectangles (including both endpoints).
- Forgetting to multiply by $\Delta x$.
Exam tipUse the Riemann lab on this site to check your hand computation and see the rectangles.
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