What is the integral from 0 to 2-2x of (2x+2y-4) ?

The integral from 0 to 2-2x of (2x+2y-4) is 4x-4

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integral from 0 to 2-2x of (2x+2y-4)

\int_{0}^{2 - 2 x} (2 x + 2 y - 4) d y

4 x - 4

Solution steps

\int_{0}^{2 - 2 x} 2 x + 2 y - 4 d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{0}^{2 - 2 x} 4 d y + \int_{0}^{2 - 2 x} 2 x d y + \int_{0}^{2 - 2 x} 2 y d y

\int_{0}^{2 - 2 x} 4 d y = 4 (- 2 x + 2)

\int_{0}^{2 - 2 x} 2 x d y = 2 x (- 2 x + 2)

\int_{0}^{2 - 2 x} 2 y d y = (- 2 x + 2)^{2}

= - 4 (- 2 x + 2) + 2 x (- 2 x + 2) + (- 2 x + 2)^{2}

Simplify

- 4 (- 2 x + 2) + 2 x (- 2 x + 2) + (- 2 x + 2)^{2} : 4 x - 4

= 4 x - 4

What is the integral from 0 to 2-2x of (2x+2y-4) ?
The integral from 0 to 2-2x of (2x+2y-4) is 4x-4