sin(x), exp(-x*y), sqrt(x), PI, and ^ (auto-converted to exponent).
How to use this multiple integration calculator
This tool estimates single, double, and triple integrals over rectangular regions. Enter a function, choose the number of variables, set the bounds, and click calculate. It is ideal for homework checks, quick numerical experiments, and validating symbolic steps.
Step-by-step
- Enter an integrand: write your function in terms of
x,y, andz. - Select dimension: choose 1, 2, or 3 integrals.
- Set bounds: lower and upper limits for each active variable.
- Choose subdivisions: larger values usually improve accuracy but take more time.
- Calculate: the result appears immediately with method details.
What is multiple integration?
Multiple integration extends the familiar area-under-a-curve idea into higher dimensions. A double integral typically computes volume under a surface over a 2D region. A triple integral extends this further to accumulate values throughout a 3D region.
Double integrals
A double integral has the form ∫∫ f(x,y) dA. In rectangular coordinates with constant bounds, this can be written as an iterated integral:
∫[x=a..b] ∫[y=c..d] f(x,y) dy dx
The calculator approximates this by summing many small rectangular cells.
Triple integrals
A triple integral has the form ∫∫∫ f(x,y,z) dV. This is useful in mass, charge, probability density, and heat distribution problems. The calculator divides the 3D box into tiny sub-volumes and adds contributions from each midpoint.
Numerical method used
This page uses the midpoint Riemann sum. For each small interval/cell/box, the function is evaluated at its center and multiplied by the tiny measure dx, dxdy, or dxdydz. The total is the integral estimate.
- More subdivisions generally increase precision.
- Smooth functions converge faster.
- Highly oscillatory or singular functions may need many more subdivisions.
Quick test examples
∫[0..1]∫[0..1] (x+y) dy dx = 1∫[0..2]∫[0..3] x*y dy dx = 9∫[0..1]∫[0..1]∫[0..1] x*y*z dz dy dx = 1/8 = 0.125
Tips for better results
Increase subdivisions gradually
Start with moderate values, then double them and compare outputs. If results stabilize, your estimate is likely reliable.
Watch domains
Functions like sqrt(x) and log(x) require valid input ranges. If any midpoint is outside the domain, the calculator warns you.
Use consistent units
If your function represents a physical density, ensure bounds and variables use consistent units so the computed integral has meaningful interpretation.
FAQ
Does it solve symbolic integrals?
No. This calculator is numerical, not symbolic. It gives approximations.
Can I use trigonometric and exponential functions?
Yes. Common Math functions like sin, cos, tan, exp, log, and constants like PI are supported.
Why does my answer differ from a textbook value?
That usually means you need more subdivisions, different variable order checks, or a closer look at bounds and expression syntax.