What is the integral from 0 to 1 of 2sqrt(1-x) ?

The integral from 0 to 1 of 2sqrt(1-x) is 4/3

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integral from 0 to 1 of 2sqrt(1-x)

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

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

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

Take the constant out:

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

= 2 \cdot \int_{0}^{1} 1 - x d x

Apply u-substitution

: \int_{1}^{0} - u d u

= 2 \cdot \int_{1}^{0} - u d u

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

= 2 (- \int_{0}^{1} - u d u)

Take the constant out:

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

= 2 (- (- \int_{0}^{1} u d u))

Apply the Power Rule

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

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

Simplify

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

Compute the boundaries:

\frac{2}{3}

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

Simplify

= \frac{4}{3}

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What is the integral from 0 to 1 of 2sqrt(1-x) ?
The integral from 0 to 1 of 2sqrt(1-x) is 4/3