What is the integral from 0 to 3 of 2pix(3-sqrt(x)) ?

The integral from 0 to 3 of 2pix(3-sqrt(x)) is 27pi-(36sqrt(3)pi)/5

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integral from 0 to 3 of 2pix(3-sqrt(x))

\int_{0}^{3} 2 π x (3 - x) d x

27 π - \frac{36 3 π}{5}

Decimal Notation

45.64493 \dots

Solution steps

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

Expand

x (3 - x) : 3 x - x^{\frac{3}{2}}

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

\int_{0}^{3} 3 x d x = \frac{27}{2}

\int_{0}^{3} x^{\frac{3}{2}} d x = \frac{18 3}{5}

= 2 π (\frac{27}{2} - \frac{18 3}{5})

Simplify

2 π (\frac{27}{2} - \frac{18 3}{5}) : 27 π - \frac{36 3 π}{5}

= 27 π - \frac{36 3 π}{5}

\frac{\partial}{\partial x} (4 ln (x y) - (x + y)^{2} + 5)

f (x) = \frac{( 3 x + 2 )}{( 3 x - 2 )}

\int (x ln (x)) d x

\int (x + 1) y^{2} \cdot sin (\frac{x y}{3}) d x

x \to 0 - lim (g (t))

What is the integral from 0 to 3 of 2pix(3-sqrt(x)) ?
The integral from 0 to 3 of 2pix(3-sqrt(x)) is 27pi-(36sqrt(3)pi)/5