integral from-1/3 to 0 of 3/4 (1-x^2)

解

\int_{- \frac{1}{3}}^{0} \frac{3}{4} (1 - x^{2}) d x

\frac{13}{54}

十進法表記

0.24074 \dots

解答ステップ

\int_{- \frac{1}{3}}^{0} \frac{3}{4} (1 - x^{2}) d x

定数を除く:

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

= \frac{3}{4} \cdot \int_{- \frac{1}{3}}^{0} 1 - x^{2} d x

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

= \frac{3}{4} (\int_{- \frac{1}{3}}^{0} 1 d x - \int_{- \frac{1}{3}}^{0} x^{2} d x)

\int_{- \frac{1}{3}}^{0} 1 d x = \frac{1}{3}

\int_{- \frac{1}{3}}^{0} x^{2} d x = \frac{1}{81}

= \frac{3}{4} (\frac{1}{3} - \frac{1}{81})

簡素化

\frac{3}{4} (\frac{1}{3} - \frac{1}{81}) : \frac{13}{54}

= \frac{13}{54}