What is the integral from-3 to 3 of (x-3)sqrt(9-x^2) ?

The integral from-3 to 3 of (x-3)sqrt(9-x^2) is -(27pi)/2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from-3 to 3 of (x-3)sqrt(9-x^2)

\int_{- 3}^{3} (x - 3) 9 - x^{2} d x

- \frac{27 π}{2}

Decimal

- 42.41150 \dots

Solution steps

\int_{- 3}^{3} (x - 3) 9 - x^{2} d x

Apply Integration By Parts

: [\frac{9}{2} (x - 3) (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) - \int \frac{9}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) d x]_{- 3}^{3}

= [\frac{9}{2} (x - 3) (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) - \int \frac{9}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) d x]_{- 3}^{3}

\int \frac{9}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) d x = \frac{9}{2} (x arcsin (\frac{x}{3}) + 9 - x^{2} - \frac{1}{27} (9 - x^{2})^{\frac{3}{2}})

= [\frac{9}{2} (x - 3) (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) - \frac{9}{2} (x arcsin (\frac{x}{3}) + 9 - x^{2} - \frac{1}{27} (9 - x^{2})^{\frac{3}{2}})]_{- 3}^{3}

Simplify

[\frac{9}{2} (x - 3) (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) - \frac{9}{2} (x arcsin (\frac{x}{3}) + 9 - x^{2} - \frac{1}{27} (9 - x^{2})^{\frac{3}{2}})]_{- 3}^{3} : [\frac{1}{12} (27 x sin (2 arcsin (\frac{x}{3})) - 162 arcsin (\frac{x}{3}) - 81 sin (2 arcsin (\frac{x}{3})) - 54 - x^{2} + 9 + 2 (- x^{2} + 9)^{\frac{3}{2}})]_{- 3}^{3}

= [\frac{1}{12} (27 x sin (2 arcsin (\frac{x}{3})) - 162 arcsin (\frac{x}{3}) - 81 sin (2 arcsin (\frac{x}{3})) - 54 - x^{2} + 9 + 2 (- x^{2} + 9)^{\frac{3}{2}})]_{- 3}^{3}

Compute the boundaries:

- \frac{27 π}{2}

= - \frac{27 π}{2}

\int_{2 x^{2}}^{1 + x^{2}} x + 2 y d y

\int_{3}^{9} x^{2} d x

\int_{1}^{6} t^{2} - 2 t - 15 d t

\int_{0}^{y} 2 z d z

\int_{0}^{1} 4^{5 x} d x

What is the integral from-3 to 3 of (x-3)sqrt(9-x^2) ?
The integral from-3 to 3 of (x-3)sqrt(9-x^2) is -(27pi)/2