Double Integral Calculator

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Double integral is a technique used in Calculus to find the area of a 2-dimensional shape in the X-Y plane.

To find the double integral, we find the definite integral step by step for each boundary value of ‘x’ and ‘y’.

Double integral calculator is the instant online tool which can find the double integral of a given function.

Example 1: Find the double integral value of the function, 2xy over the rectangle R = [1,3]x[0,1].

∫ ∫_R (2xy) dA > ∫₁³ ∫₀¹ (2xy) dy dx

First find the definite integral of ‘2xy’ treating ‘x’ as a constant, we get

∫₁³ (xy²) |₀¹dxè Evaluate xy² when ‘y’ is ‘0’ and ‘1’ and subtract them.

(x*1²) – (x*0²) = x > ∫₁³ x dx = x²/2 |₁³

When x=3 > (3²)/2 = 9/2

When x=1 > (1²)/2 =1/2

9/2 – 1/2 = 4

∫₁³ ∫₀¹ (2xy) dy dx = 4

Example 2: Find the double integral of the function, 2y + 3x over the rectangle R = [2,4]x[1,2].

∫ ∫_R (2y + 3x) dAè∫₂⁴ ∫₁² (2y + 3x) dy dx

First find the definite integral of ‘2y+3x’ treating ‘x’ as a constant, we get

∫₂⁴(y² + 3xy)|₁²dx > Evaluate y² + 3xy when ‘y’is ‘1’ and ‘2’ and subtract them.

(2² + 3x*2) - (1²+3x*1) = 4+6x – 1 – 3x = 3+3x

∫₂⁴ (3+3x)dx = 3x + 3x²/2 |₂⁴

When x=4 >3(4)+ 3(16/2)=36

When x=2 > 3(2)+ 3(4/2) = 12

36-12 = 24

Henceè∫₂⁴ ∫₁²(2y+3x)dy dx = 24

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Double Integral Calculator

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