What is the integral from 0 to 3 of pi(2/3 x)^2 ?

The integral from 0 to 3 of pi(2/3 x)^2 is 4pi

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integral from 0 to 3 of pi(2/3 x)^2

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

4 π

Decimal

12.56637 \dots

Solution steps

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

Take the constant out:

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

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

Simplify

(\frac{2}{3} x)^{2} : \frac{4}{9} x^{2}

= π \cdot \int_{0}^{3} \frac{4}{9} x^{2} d x

Take the constant out:

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

= π \frac{4}{9} \cdot \int_{0}^{3} x^{2} d x

Apply the Power Rule

: [\frac{x ^{3}}{3}]_{0}^{3}

= π \frac{4}{9} [\frac{x ^{3}}{3}]_{0}^{3}

Compute the boundaries:

9

= π \frac{4}{9} \cdot 9

Simplify

= 4 π

- 3 sin (3 t)

\frac{d}{d x} (sin^{- 2} (x))

f (x) = \frac{1 - 3 x}{4 + x}

\frac{d}{d t} (A t e^{- \frac{t}{3}})

sin (x)

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