Exercise · ZW-RIEMANN-SUMS-009 · ★☆☆ Basic
As \(n\to\infty\), what does a Riemann sum of a continuous function on \([a,b]\) converge to?
Choose one option.
Show answer
- B. The definite integral of \(f\) on \([a,b]\)
Explanation
As the partition is refined, the sum converges to the area under the curve, which is the definite integral of \(f\) on \([a,b]\).
Review the concept: Riemann sums
A Riemann sum approximates the area under a curve by dividing \([a,b]\) into \(n\) rectangles of width \(\Delta x\) and adding their areas, with the height of each rectangle taken at a point of its subinterval.
\[\begin{gathered}\sum_{i=1}^{n}f(x_i^{*})\,\Delta x\\ \Delta x=\frac{b-a}{n}\end{gathered}\]
More practice problems: riemann sums
- ZW-RIEMANN-SUMS-001 Compute the right-endpoint Riemann sum of f(x)=x on [0,2] with n=4 equal subintervals.
- ZW-RIEMANN-SUMS-002 Compute the left-endpoint Riemann sum of f(x)=x on [0,2] with n=4 equal subintervals.
- ZW-RIEMANN-SUMS-003 Compute the right-endpoint Riemann sum of f(x)=x^2 on [0,3] with n=3 equal subintervals.
- ZW-RIEMANN-SUMS-004 Compute the right-endpoint Riemann sum of f(x)=x^2 + 1 on [0,2] with n=2 equal subintervals.
- ZW-RIEMANN-SUMS-005 What is the formula for Delta x when dividing [a,b] into n equal subintervals?
- ZW-RIEMANN-SUMS-006 Compute Delta x for a Riemann sum of f on [1,5] with n=8 equal subintervals.
- ZW-RIEMANN-SUMS-007 Compute the left-endpoint Riemann sum of f(x)=4-x on [0,2] with n=2 equal subintervals.
- ZW-RIEMANN-SUMS-008 Compute the right-endpoint Riemann sum of f(x)=x^2 on [0,2] with n=4 equal subintervals.