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

The integral from 0 to 1 of 2pi(x+1)(x^2) is (7pi)/6

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

\int_{0}^{1} 2 π (x + 1) (x^{2}) d x

\frac{7 π}{6}

Decimal

3.66519 \dots

Solution steps

\int_{0}^{1} 2 π (x + 1) x^{2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 π \cdot \int_{0}^{1} (x + 1) x^{2} d x

Expand

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

= 2 π \cdot \int_{0}^{1} x^{3} + 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

= 2 π (\int_{0}^{1} x^{3} d x + \int_{0}^{1} x^{2} d x)

\int_{0}^{1} x^{3} d x = \frac{1}{4}

\int_{0}^{1} x^{2} d x = \frac{1}{3}

= 2 π (\frac{1}{4} + \frac{1}{3})

Simplify

2 π (\frac{1}{4} + \frac{1}{3}) : \frac{7 π}{6}

= \frac{7 π}{6}

What is the integral from 0 to 1 of 2pi(x+1)(x^2) ?
The integral from 0 to 1 of 2pi(x+1)(x^2) is (7pi)/6