What is the integral from-0.5 to 0.5 of sqrt(1-x^2) ?

The integral from-0.5 to 0.5 of sqrt(1-x^2) is (2pi+3sqrt(3))/(12)

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integral from-0.5 to 0.5 of sqrt(1-x^2)

\int_{- 0.5}^{0.5} 1 - x^{2} d x

\frac{2 π + 3 3}{12}

Decimal Notation

0.95661 \dots

Solution steps

\int_{- 0.5}^{0.5} 1 - x^{2} d x

Apply Trigonometric Substitution

: \int_{- \frac{π}{6}}^{\frac{π}{6}} cos^{2} (u) d u

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

Rewrite using trig identities

= \int_{- \frac{π}{6}}^{\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

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

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

\int_{- \frac{π}{6}}^{\frac{π}{6}} 1 d u = \frac{π}{3}

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

= \frac{1}{2} (\frac{π}{3} + \frac{3}{2})

Simplify

\frac{1}{2} (\frac{π}{3} + \frac{3}{2}) : \frac{2 π + 3 3}{12}

= \frac{2 π + 3 3}{12}

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What is the integral from-0.5 to 0.5 of sqrt(1-x^2) ?
The integral from-0.5 to 0.5 of sqrt(1-x^2) is (2pi+3sqrt(3))/(12)