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

The integral from 0 to 3 of x^2(1-x) is -45/4

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integral from 0 to 3 of x^2(1-x)

\int_{0}^{3} x^{2} (1 - x) d x

- \frac{45}{4}

Decimal

- 11.25

Solution steps

\int_{0}^{3} x^{2} (1 - x) d x

Expand

x^{2} (1 - x) : x^{2} - x^{3}

= \int_{0}^{3} x^{2} - x^{3} 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}^{3} x^{2} d x - \int_{0}^{3} x^{3} d x

\int_{0}^{3} x^{2} d x = 9

\int_{0}^{3} x^{3} d x = \frac{81}{4}

= 9 - \frac{81}{4}

Simplify

9 - \frac{81}{4} : - \frac{45}{4}

= - \frac{45}{4}

What is the integral from 0 to 3 of x^2(1-x) ?
The integral from 0 to 3 of x^2(1-x) is -45/4