What is the integral from-3 to 3 of (x+6)sqrt(9-x^2) ?

The integral from-3 to 3 of (x+6)sqrt(9-x^2) is 27pi

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integral from-3 to 3 of (x+6)sqrt(9-x^2)

\int_{- 3}^{3} (x + 6) 9 - x^{2} d x

27 π

Decimal

84.82300 \dots

Solution steps

\int_{- 3}^{3} (x + 6) 9 - x^{2} d x

Expand

(x + 6) 9 - x^{2} : x 9 - x^{2} + 6 9 - x^{2}

= \int_{- 3}^{3} x 9 - x^{2} + 6 9 - x^{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

= \int_{- 3}^{3} x 9 - x^{2} d x + \int_{- 3}^{3} 6 9 - x^{2} d x

\int_{- 3}^{3} x 9 - x^{2} d x = 0

\int_{- 3}^{3} 6 9 - x^{2} d x = 27 π

= 0 + 27 π

Simplify

= 27 π

What is the integral from-3 to 3 of (x+6)sqrt(9-x^2) ?
The integral from-3 to 3 of (x+6)sqrt(9-x^2) is 27pi