Multiple Integral Calculator
Evaluate Double (Volume) and Triple Integrals over rectangular regions.
Use x, y (and z for triple). Ex: sin(x)*y
Integration Limits (Constant)
Calculated Value
Region Visualization (Double Only)
Drag to RotateUnderstanding Multiple Integrals
Just as a single integral ∫ f(x) dx calculates the area under a curve, multiple integrals extend this concept to higher dimensions. They are fundamental tools in engineering and physics for calculating volume, mass, center of gravity, and moments of inertia.
Double Integrals (∫∫)
A double integral calculates the Volume under the surface z = f(x,y) and above a specific region on the x-y plane.
Volume = ∫∫ f(x,y) dA
Triple Integrals (∫∫∫)
A triple integral sums up values over a 3D block of space. If the function is 1, it calculates geometric volume. If the function is density ρ(x,y,z), it calculates Total Mass.
Mass = ∫∫∫ ρ(x,y,z) dV
Geometric Interpretation
Imagine cutting a potato into tiny french fries. A triple integral adds up the density of every tiny fry to give you the total weight of the potato. A double integral is like standing on a rug (the region R) and measuring the volume of the air between the rug and a curved roof (the function f).
Frequently Asked Questions (FAQ)
Q: Does the order of integration matter (dx dy vs dy dx)?
A: For continuous functions on rectangular regions, No. Fubini’s Theorem states you can switch the order and get the same result. However, for complex shaped regions, changing the order changes the limits of integration.
Q: What if the result is negative?
A: Just like single integrals, if the function drops below the axis (or plane), it counts as “negative volume”. The result is the net volume (volume above minus volume below).
Q: Can I use this for non-rectangular regions?
A: This specific tool is optimized for constant limits (rectangular boxes). For triangles or circles, you would need to set variable limits (e.g., integrate from 0 to x), which is more advanced.