What is the integral from 0 to 1/2 of xsqrt(1-3x^2) ?

The integral from 0 to 1/2 of xsqrt(1-3x^2) is 7/72

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integral from 0 to 1/2 of xsqrt(1-3x^2)

\int_{0}^{\frac{1}{2}} x 1 - 3 x^{2} d x

\frac{7}{72}

Decimal

0.09722 \dots

Solution steps

\int_{0}^{\frac{1}{2}} x 1 - 3 x^{2} d x

Apply u-substitution

= \int_{1}^{\frac{1}{4}} - \frac{u}{6} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{4}}^{1} - \frac{u}{6} d u

Take the constant out:

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

= - (- \frac{1}{6} \cdot \int_{\frac{1}{4}}^{1} u d u)

Apply the Power Rule

= - (- \frac{1}{6} [\frac{2}{3} u^{\frac{3}{2}}]_{\frac{1}{4}}^{1})

Simplify

= \frac{1}{6} [\frac{2}{3} u^{\frac{3}{2}}]_{\frac{1}{4}}^{1}

Compute the boundaries:

\frac{7}{12}

= \frac{1}{6} \cdot \frac{7}{12}

Simplify

= \frac{7}{72}

What is the integral from 0 to 1/2 of xsqrt(1-3x^2) ?
The integral from 0 to 1/2 of xsqrt(1-3x^2) is 7/72