Calculator
Example Data Table
| Function | Interval | Partition | Method | Notes |
|---|---|---|---|---|
| x^2 + 1 | [0, 4] | 8 equal parts | Midpoint | Good basic polynomial test. |
| sin(x) | [0, pi] | 12 equal parts | Trapezoid | Use radians for this example. |
| 1/(1+x^2) | [0, 2] | 0, .25, .5, 1, 2 | Right endpoint | Shows uneven partition widths. |
Formula Used
For a partition P = {x0, x1, ..., xn}, each width is Δxi = xi - xi-1.
The general Riemann sum is Σ f(ci) Δxi, where ci is the selected sample point.
Left sum uses ci = xi-1. Right sum uses ci = xi. Midpoint sum uses ci = (xi-1 + xi) / 2.
Custom fraction uses ci = xi-1 + αΔxi. Trapezoid sum uses Σ ((f(xi-1) + f(xi)) / 2) Δxi.
How to Use This Calculator
Enter the function in terms of x. Choose a uniform or custom partition. For a uniform partition, enter a, b, and n. For a custom partition, enter ordered points.
Select the method. Use left, right, midpoint, trapezoid, or a custom sample fraction. Add an exact value if you want an error check. Press calculate. Review the result table above the form. Then export CSV or PDF.
Riemann Sums With Partitions
A Riemann sum estimates area under a curve. It breaks an interval into smaller parts. Each part has a width. A sample point is chosen inside that part. The calculator evaluates the function there. Then it multiplies height by width. Finally, it adds every small signed area. Use the notes to compare different partition choices clearly before making a final class submission.
Why partitions matter
A partition controls the width of each slice. Equal partitions are simple and fast. Custom partitions are useful when the curve changes quickly in one region. Smaller widths often improve the estimate. Uneven widths can focus work where more detail is needed. That makes the result more flexible than a fixed grid.
Available sum methods
The left method uses each left endpoint. The right method uses each right endpoint. The midpoint method samples the center of every slice. The custom fraction method samples a point between endpoints. The trapezoid method averages two endpoint heights. Each method can behave differently on increasing, decreasing, or curved functions.
Reading the output
The result table shows each subinterval. It lists the starting point, ending point, width, sample point, function value, and area term. This makes the estimate easy to audit. It also helps students see how every rectangle or trapezoid contributes to the final total.
Accuracy notes
A Riemann sum is an approximation unless the function and partition create an exact area. Increasing the number of subintervals usually improves accuracy for smooth functions. The calculator also accepts a known exact value. When you enter it, the tool reports absolute and relative error.
Good inputs
Use x as the variable. Enter expressions such as x^2, sin(x), sqrt(x), exp(x), or 1/(1+x^2). Choose radians or degrees for trigonometric work. For custom partitions, enter ordered values separated by commas or spaces. Check the partition table before exporting results.
Common uses
Teachers use these sums to explain definite integrals. Students use them to check homework steps. Analysts can approximate area, accumulated change, distance, and total production. Engineers may use partitions when a measured curve has changing loads. The export buttons save the table for reports, worksheets, or later review. Keep more decimal places when small widths create tiny area terms.
FAQs
What is a Riemann sum?
It is an estimate of area under a curve. The interval is split into parts. Each part uses a function height and a width. The products are added.
What is a partition?
A partition is the list of points that divides an interval. Consecutive points form subintervals. Each subinterval has its own width.
Which method should I use?
Use midpoint for many classroom estimates. Use left or right when a problem asks for them. Use trapezoid when you want endpoint averaging.
Can I use unequal widths?
Yes. Select custom partition. Then enter ordered points separated by commas, spaces, or lines. The calculator uses each actual width.
Does the calculator support trigonometric functions?
Yes. It supports sin, cos, tan, and inverse trig functions. Choose radians or degrees before calculating the result.
Why is my result negative?
The calculator returns signed area. A negative value can occur below the x-axis or when the interval direction is reversed.
How do I check error?
Enter the known exact integral value. The calculator will show absolute error and relative error when that value is available.
Can I export the partition table?
Yes. After calculation, use the CSV button for spreadsheet work. Use the PDF button for a quick printable report.