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

The integral from 0 to 2 of (-x^2+4x-x^2) is 8/3

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

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

\frac{8}{3}

Decimal Notation

2.66666 \dots

Solution steps

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

Simplify

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

\int_{0}^{2} 2 x^{2} d x = \frac{16}{3}

\int_{0}^{2} 4 x d x = 8

= - \frac{16}{3} + 8

Simplify

- \frac{16}{3} + 8 : \frac{8}{3}

= \frac{8}{3}

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What is the integral from 0 to 2 of (-x^2+4x-x^2) ?
The integral from 0 to 2 of (-x^2+4x-x^2) is 8/3