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Mathematics · University

A separable double integral

The rectangular region allows two independent single-variable integrations.

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Method and conditions

The rectangular region allows two independent single-variable integrations.

∫₀ᵃ∫₀ᵇ kxy dy dx = ka²b²/4

Worked examples

Integrate 9xy over the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 3. Round the final result to 4 decimal places if needed.

Integrate y first to get 9x(3²/2), then integrate x over [0,1].

20.25

Integrate 3xy over the rectangle 0 ≤ x ≤ 8, 0 ≤ y ≤ 5. Round the final result to 4 decimal places if needed.

Integrate y first to get 3x(5²/2), then integrate x over [0,8].

1200

Try it

  1. Integrate 5xy over the rectangle 0 ≤ x ≤ 10, 0 ≤ y ≤ 9. Round the final result to 4 decimal places if needed.

  2. Integrate 8xy over the rectangle 0 ≤ x ≤ 5, 0 ≤ y ≤ 9. Round the final result to 4 decimal places if needed.

Check your work

Answers and marking notes
  1. 10125

    Integrate y first to get 5x(9²/2), then integrate x over [0,10].

  2. 4050

    Integrate y first to get 8x(9²/2), then integrate x over [0,5].

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