- Estimate the area under a curve by adding up the areas of rectangles
- Estimate the area under a curve using Riemann sums
Approximating Area
The Main Idea
- Area under a curve can be approximated using rectangles
- Two main methods: left-endpoint and right-endpoint approximations
- Increasing the number of rectangles improves accuracy
- Riemann sums formalize this approximation process
- Partitions:
- Divide interval [latex][a,b][/latex] into n subintervals
- Regular partition: subintervals of equal width [latex]\Delta x = \frac{(b-a)}{n}[/latex]
- Left-Endpoint Approximation:
- [latex]L_n = \sum_{i=1}^n f(x_{i-1})\Delta x[/latex]
- Rectangle heights based on function value at left endpoint
- Right-Endpoint Approximation:
- [latex]R_n = \sum_{i=1}^n f(x_i)\Delta x[/latex]
- Rectangle heights based on function value at right endpoint
- Convergence:
- As [latex]n[/latex] increases, approximations generally become more accurate
- Left and right approximations often converge to the true area
Find the sum of the values of [latex]f(x)=x^3[/latex] over the integers [latex]1,2,3,\cdots,10[/latex].
Find left and right-endpoint approximations for [latex]\int_0^2 x^2 dx[/latex] with [latex]n = 4[/latex].
Riemann Sums
The Main Idea
- Riemann sums generalize left and right endpoint approximations. They allow evaluation of the function at any point within each subinterval
- As the number of subintervals increases, the approximation improves
- Riemann sums can be used to define the area under a curve
- Riemann Sum Definition:
- [latex]\displaystyle\sum_{i=1}^{n} f(x_i^*)\Delta x[/latex] where [latex]x_i^*[/latex] is any point in the subinterval [latex][x_{i-1},x_i][/latex]
- Area Under a Curve:
- [latex]A=\underset{n\to \infty }{\lim}\displaystyle\sum_{i=1}^{n} f(x_i^*)\Delta x[/latex]
- Upper and Lower Sums:
- Upper sum:
- Choose [latex]x_i^*[/latex] for maximum [latex]f(x_i^*)[/latex] in each subinterval
- Lower sum:
- Choose [latex]x_i^*[/latex] for minimum [latex]f(x_i^*)[/latex] in each subinterval
- Upper sum:
- Convergence:
- For continuous functions, the limit of Riemann sums is unique
- The limit doesn’t depend on the choice of [latex]x_i^*[/latex]
Find a lower sum for [latex]f(x)= \sin x[/latex] over the interval [latex][a,b]=[0,\frac{\pi }{2}][/latex]; let [latex]n=6[/latex].
Using the function [latex]f(x)= \sin x[/latex] over the interval [latex][0,\frac{\pi}{2}][/latex], find an upper sum; let [latex]n=6[/latex].