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

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

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

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

\frac{27 π}{2}

Decimal

42.41150 \dots

Solution steps

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

Take the constant out:

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

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

Apply Trigonometric Substitution

= 3 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 9 cos^{2} (u) d u

Take the constant out:

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

= 3 \cdot 9 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{2} (u) d u

Rewrite using trig identities

= 3 \cdot 9 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= 3 \cdot 9 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + cos (2 u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 \cdot 9 \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d u + \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 u) d u)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d u = π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 u) d u = 0

= 3 \cdot 9 \cdot \frac{1}{2} (π + 0)

Simplify

3 \cdot 9 \cdot \frac{1}{2} (π + 0) : \frac{27 π}{2}

= \frac{27 π}{2}

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