integral de 0 a 2 de sqrt(4x-x^2)

Solución

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

π

Pasos de solución

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

Completar el cuadrado

4 x - x^{2} : - (x - 2)^{2} + 4

= \int_{0}^{2} - (x - 2)^{2} + 4 d x

Aplicar integración por sustitución

: \int_{- 2}^{0} - u^{2} + 4 d u

= \int_{- 2}^{0} - u^{2} + 4 d u

Aplicar integración por sustitución trigonométrica

: \int_{- \frac{π}{2}}^{0} 4 cos^{2} (v) d v

= \int_{- \frac{π}{2}}^{0} 4 cos^{2} (v) d v

Sacar la constante:

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

= 4 \cdot \int_{- \frac{π}{2}}^{0} cos^{2} (v) d v

Re-escribir usando identidades trigonométricas

= 4 \cdot \int_{- \frac{π}{2}}^{0} \frac{1 + cos ( 2 v )}{2} d v

Sacar la constante:

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

= 4 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{0} 1 + cos (2 v) d v

Aplicar la regla de la suma:

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

= 4 \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{0} 1 d v + \int_{- \frac{π}{2}}^{0} cos (2 v) d v)

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

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

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

Simplificar

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

= π

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