What is the integral from 1/3 to 2/3 of sqrt(4-9x^2) ?

The integral from 1/3 to 2/3 of sqrt(4-9x^2) is (4pi-3sqrt(3))/(18)

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integral from 1/3 to 2/3 of sqrt(4-9x^2)

\int_{\frac{1}{3}}^{\frac{2}{3}} 4 - 9 x^{2} d x

\frac{4 π - 3 3}{18}

Decimal

0.40945 \dots

Solution steps

\int_{\frac{1}{3}}^{\frac{2}{3}} 4 - 9 x^{2} d x

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

= \frac{4}{3} \cdot \int_{\frac{π}{6}}^{\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

= \frac{4}{3} \cdot \frac{1}{2} \cdot \int_{\frac{π}{6}}^{\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

= \frac{4}{3} \cdot \frac{1}{2} (\int_{\frac{π}{6}}^{\frac{π}{2}} 1 d u + \int_{\frac{π}{6}}^{\frac{π}{2}} cos (2 u) d u)

\int_{\frac{π}{6}}^{\frac{π}{2}} 1 d u = \frac{π}{3}

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

= \frac{4}{3} \cdot \frac{1}{2} (\frac{π}{3} - \frac{3}{4})

Simplify

\frac{4}{3} \cdot \frac{1}{2} (\frac{π}{3} - \frac{3}{4}) : \frac{4 π - 3 3}{18}

= \frac{4 π - 3 3}{18}

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What is the integral from 1/3 to 2/3 of sqrt(4-9x^2) ?
The integral from 1/3 to 2/3 of sqrt(4-9x^2) is (4pi-3sqrt(3))/(18)