integral from-10 to 10 of sqrt(100-x^2)

Lösung

\int_{- 10}^{10} 100 - x^{2} d x

50 π

Dezimale

157.07963 \dots

Schritte zur Lösung

\int_{- 10}^{10} 100 - x^{2} d x

Trigonometrische Substitution anwenden

= \int_{- \frac{π}{2}}^{\frac{π}{2}} 100 cos^{2} (u) d u

Entferne die Konstante:

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

= 100 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 100 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + cos ( 2 u )}{2} d u

Entferne die Konstante:

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

= 100 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + cos (2 u) d u

Wende die Summenregel an:

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

= 100 \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d u + \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 u) d u)

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

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

= 100 \cdot \frac{1}{2} (π + 0)

Vereinfache

100 \cdot \frac{1}{2} (π + 0) : 50 π

= 50 π

cos (y) y^{^{'}} + \frac{1}{x} sin (y) = 1

a re a y = 2 x, y = 2, x = 0

y^{^{'}} = \frac{y ^{3}}{e ^{x}}, y (0) = - 1

\frac{d y}{d x} = (4 - y)^{5}

(1 + x^{2}) \frac{d y}{d x} = 2 x y + x^{2} + x^{4}