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MathidiaMathematics Academy
Multivariable and vector calculus
Lesson 6 of 75
Introduction

Double integrals

Objective. Change order and region descriptions in double integrals.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Double integrals compute the signed volume under a surface z = f(x, y) over a planar region R, evaluated by iterated single integrals once limits are expressed correctly for the region's shape. For a type-I region where y ranges between curves g₁(x) and g₂(x) over x ∈ [a, b], ∬_R f dA = ∫ₐᵇ ∫_{g₁(x)}^{g₂(x)} f(x,y) dy dx. Swapping the order requires rewriting the region as type II, bounded by functions of y. For f(x, y) = xy over the unit square [0,1]×[0,1], the integral splits as (∫₀¹ x dx)(∫₀¹ y dy) = (1/2)(1/2) = 1/4. Correctly identifying the region's boundary curves is essential before integrating.

Worked example

Evaluate the double integral ∫₀¹ ∫₀² xy dy dx.

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Illustrated card showing ?
  1. 1Integrate the inner integral with respect to y, treating x as constant: ∫₀² xy dy = x[y²/2]₀² = 2x.
  2. 2The double integral becomes ∫₀¹ 2x dx.
  3. 3Integrate: [x²]₀¹.
  4. 4Value = 1 − 0 = 1.

Answer: 1

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Common mistake. A frequent mistake is using constant limits for both variables even when the region is not rectangular, which integrates over the wrong area; always sketch the region first to find variable bounds.

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