What is the integral from 0 to 9/2 of sqrt(81-x^2) ?

The integral from 0 to 9/2 of sqrt(81-x^2) is (27(2pi+3sqrt(3)))/8

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integral from 0 to 9/2 of sqrt(81-x^2)

\int_{0}^{\frac{9}{2}} 81 - x^{2} d x

\frac{27 ( 2 π + 3 3 )}{8}

Decimal

38.74276 \dots

Solution steps

\int_{0}^{\frac{9}{2}} 81 - x^{2} d x

Apply Trigonometric Substitution

= \int_{0}^{\frac{π}{6}} 81 cos^{2} (u) d u

Take the constant out:

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

= 81 \cdot \int_{0}^{\frac{π}{6}} cos^{2} (u) d u

Rewrite using trig identities

= 81 \cdot \int_{0}^{\frac{π}{6}} \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

= 81 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{6}} 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

= 81 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{6}} 1 d u + \int_{0}^{\frac{π}{6}} cos (2 u) d u)

\int_{0}^{\frac{π}{6}} 1 d u = \frac{π}{6}

\int_{0}^{\frac{π}{6}} cos (2 u) d u = \frac{3}{4}

= 81 \cdot \frac{1}{2} (\frac{π}{6} + \frac{3}{4})

Simplify

81 \cdot \frac{1}{2} (\frac{π}{6} + \frac{3}{4}) : \frac{27 ( 2 π + 3 3 )}{8}

= \frac{27 ( 2 π + 3 3 )}{8}

\frac{d x}{d t} = 0.6 - \frac{3 x}{500}

\frac{d y}{d x} = 8 x (y - 3)

\int \frac{3}{4} x^{2} - 5 x^{2} d x

y^{^{'}} = - y^{2} + 6

\int \frac{x ^{2}}{x ^{2} - 2} d x

What is the integral from 0 to 9/2 of sqrt(81-x^2) ?
The integral from 0 to 9/2 of sqrt(81-x^2) is (27(2pi+3sqrt(3)))/8