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

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

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

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

3 π

Decimal

5.44139 \dots

Solution steps

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

Apply Integration By Parts

: [\frac{1}{2} x 12 - 3 x^{2} - \int - \frac{3 x ^{2}}{2 12 - 3 x ^{2}} d x]_{- 2}^{2}

= [\frac{1}{2} x 12 - 3 x^{2} - \int - \frac{3 x ^{2}}{2 12 - 3 x ^{2}} d x]_{- 2}^{2}

\int - \frac{3 x ^{2}}{2 12 - 3 x ^{2}} d x = - 3 (arcsin (\frac{x}{2}) - \frac{1}{2} sin (2 arcsin (\frac{x}{2})))

= [\frac{1}{2} x 12 - 3 x^{2} - (- 3 (arcsin (\frac{x}{2}) - \frac{1}{2} sin (2 arcsin (\frac{x}{2}))))]_{- 2}^{2}

Simplify

= [\frac{1}{2} x 12 - 3 x^{2} + 3 (arcsin (\frac{x}{2}) - \frac{1}{2} sin (2 arcsin (\frac{x}{2})))]_{- 2}^{2}

Compute the boundaries:

3 π

= 3 π

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