Double Integrals

Key idea

Volume under a surface over a region.

In depth

∬ f dA sums f over a 2D region. Evaluate as an iterated integral, integrating one variable at a time with correct limits. It gives volume under a surface, or area when f = 1.

Worked example

∫₀¹∫₀² (x + y) dx dy = ∫₀¹ (2 + 2y) dy = 3.

Formula

Real-world uses

• Calculating the total mass of a flat plate or the volume under a hilly surface

Common mistake

Switching the order incorrectly when limits depend on a variable.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Calculus

• Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation – Take logs before differentiating products and powers.
• Related Rates – Link rates of change of connected quantities.
• Linear Approximation – Approximate f near a using its tangent line.
• Differentials – Use dy = f′(x)dx for small changes.