The shaded bell interval drives a cumulative-integral trace below itLive current inputs
Math & Statistics
Integral Graphing Calculator
Graph a Gaussian-shaped signal above a baseline, shade a selected interval, and connect the local curve to its cumulative analytic integral using the error function.
Shaded signed area -
Gaussian-only area -
Baseline contribution -
Full Gaussian area -
Gaussian area captured -
Average value in interval -
Lower standardized bound -
Upper standardized bound -
AREA AND ACCUMULATION
The shaded bell interval drives a cumulative-integral trace below it
The first panel shows the actual integrand and selected area; the second shows how the integral accumulates from the left display edge to each x.
BOUNDARY CHECKPOINTS
Function height and accumulated area at meaningful locations
The checkpoints include display edges, selected bounds, the center, and the cumulative endpoint so local height is not confused with area.
| Checkpoint | x | Standardized z | Function value | Cumulative from display left |
|---|
GRAPH SETUP
Use the same scale for the curve, interval, and cumulative story
- Enter amplitude and baseline with their signs intact.
- Enter a strictly positive sigma.
- Place lower and upper bounds around the interval of interest.
- Extend the display span when the selected bounds approach the graph edge.
- Compare captured Gaussian area with the full Gaussian area before calling the interval complete.
HEIGHT IS NOT AREA
A narrow high peak can integrate to less than a broad modest peak
Amplitude controls height while sigma controls width. The full Gaussian-only area is their product times sigma and square root of two pi.
The cumulative curve is monotone only when the full integrand is nonnegative. A negative amplitude or baseline can produce declining sections, which the signed model preserves.
GAUSSIAN ANTIDERIVATIVE
Separate the constant baseline from the bell-shaped contribution
The baseline integrates to a rectangle. The Gaussian term integrates through erf after standardizing distance from the center by sigma times square root of two.
Detailed calculation process and general formulas
f(x)=B+A exp(-(x-mu)^2/(2 sigma^2))I_B=B(u-l)I_G=A sigma sqrt(pi/2)[erf(z_u)-erf(z_l)]z_x=(x-mu)/(sqrt(2)sigma)I=I_B+I_GSymbols, meanings, and units
- A
- Gaussian peak amplitude above baselinef-unit
- B
- constant baselinef-unit
- mu
- Gaussian centerx-unit
- sigma
- Gaussian width parameterx-unit
- erf
- error functiondimensionless
- I
- signed area between the selected boundsf-unit*x-unit
INTERVAL INTERPRETATION
Three views of what the selected window contains
The same shaded interval can be described as bell mass, baseline exposure, or average level.
Bell contribution
-The Gaussian-only area responds to center and width.
Baseline contribution
-The constant part scales directly with interval width.
Captured share
-The interval bell area is compared with the full Gaussian bell area.
Decision takeaway: Report which contribution is meaningful instead of treating every area under a baseline-plus-peak curve as one phenomenon.
CURVE FIT NOTES
Information needed before using the Gaussian as a measured model
- How A, mu, sigma, and baseline were estimated
- Whether the peak is symmetric in observed data
- Units and calibration of both axes
- Whether tails outside the display window matter
- Whether overlapping peaks were omitted
Applied decisions
Why the cumulative panel changes interpretation
Sensor pulse
A transient Gaussian-like signal rides above a constant background.
What the result clarifies: The area split keeps pulse exposure separate from baseline exposure.
Concentration peak
A selected sampling window captures most but not all of a symmetric peak.
What the result clarifies: Captured share quantifies tail loss rather than relying on a visual guess.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This calculator integrates the entered Gaussian-plus-baseline model. Real peaks can be skewed, multimodal, truncated, or correlated with a changing baseline; those structures require a different model.
Integral Graphing Calculator | Gaussian Area and Cumulative Integral FAQ
Why use erf?
The Gaussian has no elementary antiderivative; its definite integral is conventionally expressed with the error function.
What does captured percentage exclude?
It compares only the Gaussian contribution, not the constant baseline.
Can amplitude be negative?
Yes. The page then represents a Gaussian-shaped dip and preserves signed area.
Why show cumulative area?
It reveals where the total is accumulated rather than only the final endpoint difference.