Multiple IntegralsMedium

Double integral over a rectangle

Calculate

\displaystyle\iint_D (x+y)\,dA

on D = [0,1]×[0,2].

Given data

D = [0,1]×[0,2]

Steps

0 / 2

Integrate with respect to y with x fixed: ∫₀²(x+y) dy.
Write your working — the numerical result must appear in the text (logica)
Integrate with respect to x: ∫₀¹(2x+2) dx.

Full worked solution

Integrate with respect to y with x fixed: ∫₀²(x+y) dy.
$\int_0^2(x+y)\,dy = 2x+2$
$\int_0^2 (x+y)\,dy = \left[xy + \frac{y^2}{2}\right]_0^2 = 2x + 2$ . First integrate in $y$ , treating $x$ as constant.
Integrate with respect to x: ∫₀¹(2x+2) dx.
$\int_0^1(2x+2)\,dx = 3$
$\int_0^1 (2x+2)\,dx = \left[x^2 + 2x\right]_0^1 = 1 + 2 = \mathbf{3}$ . Integrating the result in $x$ gives the final value of the double integral.

Result:∬(x+y) dA = 3.

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