What is the integral from 0 to 1/2 of xsqrt(1-4x^2) ?

The integral from 0 to 1/2 of xsqrt(1-4x^2) is 1/12

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integral from 0 to 1/2 of xsqrt(1-4x^2)

\int_{0}^{\frac{1}{2}} x 1 - 4 x^{2} d x

\frac{1}{12}

Decimal

0.08333 \dots

Solution steps

\int_{0}^{\frac{1}{2}} x 1 - 4 x^{2} d x

Apply u-substitution

= \int_{1}^{0} - \frac{u}{8} d u

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

= - \int_{0}^{1} - \frac{u}{8} d u

Take the constant out:

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

= - (- \frac{1}{8} \cdot \int_{0}^{1} u d u)

Apply the Power Rule

= - (- \frac{1}{8} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1})

Simplify

= \frac{1}{8} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}

Compute the boundaries:

\frac{2}{3}

= \frac{1}{8} \cdot \frac{2}{3}

Simplify

= \frac{1}{12}

What is the integral from 0 to 1/2 of xsqrt(1-4x^2) ?
The integral from 0 to 1/2 of xsqrt(1-4x^2) is 1/12