What is the integral from 0 to 1 of 2pi*2/3*x*2/3 ?

The integral from 0 to 1 of 2pi*2/3*x*2/3 is (4pi)/9

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integral from 0 to 1 of 2pi2/3x*2/3

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

\frac{4 π}{9}

Decimal Notation

1.39626 \dots

Solution steps

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

Take the constant out:

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

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

2 π \frac{2}{3} \cdot \frac{2}{3} = \frac{8 π}{9}

= \frac{8 π}{9} \cdot \int_{0}^{1} x d x

Apply the Power Rule

: [\frac{x ^{2}}{2}]_{0}^{1}

= \frac{8 π}{9} [\frac{x ^{2}}{2}]_{0}^{1}

Compute the boundaries:

\frac{1}{2}

= \frac{8 π}{9} \cdot \frac{1}{2}

Simplify

= \frac{4 π}{9}

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y = \frac{4}{1 + x ^{2}}, y = 1

\frac{d}{d t} (t^{2} - 2 t)

What is the integral from 0 to 1 of 2pi*2/3*x*2/3 ?
The integral from 0 to 1 of 2pi*2/3*x*2/3 is (4pi)/9