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

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

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integral from 0 to 2 of (1/2 x+1/2)-x^2

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

- \frac{2}{3}

Decimal

- 0.66666 \dots

Solution steps

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

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

\int_{0}^{2} \frac{1}{2} x d x = 1

\int_{0}^{2} \frac{1}{2} d x = 1

\int_{0}^{2} x^{2} d x = \frac{8}{3}

= 1 + 1 - \frac{8}{3}

Simplify

1 + 1 - \frac{8}{3} : - \frac{2}{3}

= - \frac{2}{3}

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