What is the integral from 0 to 1-x of x^2+y^2 ?

The integral from 0 to 1-x of x^2+y^2 is x^2(-x+1)+((-x+1)^3)/3

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integral from 0 to 1-x of x^2+y^2

\int_{0}^{1 - x} x^{2} + y^{2} d y

x^{2} (- x + 1) + \frac{( - x + 1 ) ^{3}}{3}

Solution steps

\int_{0}^{1 - x} x^{2} + y^{2} 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}^{1 - x} x^{2} d y + \int_{0}^{1 - x} y^{2} d y

\int_{0}^{1 - x} x^{2} d y = x^{2} (- x + 1)

\int_{0}^{1 - x} y^{2} d y = \frac{( - x + 1 ) ^{3}}{3}

= x^{2} (- x + 1) + \frac{( - x + 1 ) ^{3}}{3}

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What is the integral from 0 to 1-x of x^2+y^2 ?
The integral from 0 to 1-x of x^2+y^2 is x^2(-x+1)+((-x+1)^3)/3