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

The integral from-1 to 1 of (1-x)sqrt(1-x^2) is pi/2

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

\int_{- 1}^{1} (1 - x) 1 - x^{2} d x

\frac{π}{2}

Decimal

1.57079 \dots

Solution steps

\int_{- 1}^{1} (1 - x) 1 - x^{2} d x

Expand

(1 - x) 1 - x^{2} : 1 - x^{2} - x 1 - x^{2}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int_{- 1}^{1} 1 - x^{2} d x = \frac{π}{2}

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

= \frac{π}{2} - 0

Simplify

= \frac{π}{2}

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What is the integral from-1 to 1 of (1-x)sqrt(1-x^2) ?
The integral from-1 to 1 of (1-x)sqrt(1-x^2) is pi/2