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

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

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

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

3 π

Decimal

9.42477 \dots

Solution steps

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

Apply Integration By Parts

: [\frac{3}{2} x 4 - x^{2} - \int - \frac{3 x ^{2}}{2 - x ^{2} + 4} d x]_{- 2}^{2}

= [\frac{3}{2} x 4 - x^{2} - \int - \frac{3 x ^{2}}{2 - x ^{2} + 4} d x]_{- 2}^{2}

\int - \frac{3 x ^{2}}{2 - x ^{2} + 4} d x = - 3 (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x)))

= [\frac{3}{2} x 4 - x^{2} - (- 3 (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x))))]_{- 2}^{2}

Simplify

= [\frac{3}{2} x 4 - x^{2} + 3 (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x)))]_{- 2}^{2}

Compute the boundaries:

3 π

= 3 π

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