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

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

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

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

\frac{π}{4}

Decimal

0.78539 \dots

Solution steps

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

Apply Trigonometric Substitution

: \int_{- \frac{π}{2}}^{0} cos^{2} (u) d u

= \int_{- \frac{π}{2}}^{0} cos^{2} (u) d u

Rewrite using trig identities

= \int_{- \frac{π}{2}}^{0} \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{0} 1 + cos (2 u) d u

Apply the Sum Rule:

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

= \frac{1}{2} (\int_{- \frac{π}{2}}^{0} 1 d u + \int_{- \frac{π}{2}}^{0} cos (2 u) d u)

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

\int_{- \frac{π}{2}}^{0} cos (2 u) d u = 0

= \frac{1}{2} (\frac{π}{2} + 0)

Simplify

\frac{1}{2} (\frac{π}{2} + 0) : \frac{π}{4}

= \frac{π}{4}

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\int_{- 1}^{1} (3 x - 9) d x

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