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人気のある微分積分 >

integral from-10 to 0 of sqrt(100-x^2)

解

\int_{- 10}^{0} 100 - x^{2} d x

解

25 π

+1

十進法表記

78.53981 \dots

解答ステップ

\int_{- 10}^{0} 100 - x^{2} d x

三角関数による置換を適用する

= \int_{- \frac{π}{2}}^{0} 100 cos^{2} (u) d u

定数を除く:

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

= 100 \cdot \int_{- \frac{π}{2}}^{0} cos^{2} (u) d u

三角関数の公式を使用して書き換える

= 100 \cdot \int_{- \frac{π}{2}}^{0} \frac{1 + cos ( 2 u )}{2} d u

定数を除く:

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

= 100 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{0} 1 + cos (2 u) d u

総和規則を適用する:

\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}}^{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

= 100 \cdot \frac{1}{2} (\frac{π}{2} + 0)

簡素化

100 \cdot \frac{1}{2} (\frac{π}{2} + 0) : 25 π

= 25 π

グラフ

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