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

The integral from 0 to 2 of 2pix(2x^2-x^3) is (16pi)/5

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{16 π}{5}

Decimal

10.05309 \dots

Solution steps

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

Expand

x (2 x^{2} - x^{3}) : 2 x^{3} - x^{4}

= 2 π \cdot \int_{0}^{2} 2 x^{3} - x^{4} 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}^{2} 2 x^{3} d x - \int_{0}^{2} x^{4} d x)

\int_{0}^{2} 2 x^{3} d x = 8

\int_{0}^{2} x^{4} d x = \frac{32}{5}

= 2 π (8 - \frac{32}{5})

Simplify

2 π (8 - \frac{32}{5}) : \frac{16 π}{5}

= \frac{16 π}{5}

What is the integral from 0 to 2 of 2pix(2x^2-x^3) ?
The integral from 0 to 2 of 2pix(2x^2-x^3) is (16pi)/5