integral from-r to r of pi(r^2-x^2)

解

\int_{- r}^{r} π (r^{2} - x^{2}) d x

π (2 r^{3} - \frac{2 r ^{3}}{3})

解答ステップ

\int_{- r}^{r} π (r^{2} - x^{2}) d x

定数を除く:

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

= π \cdot \int_{- r}^{r} r^{2} - 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

= π (\int_{- r}^{r} r^{2} d x - \int_{- r}^{r} x^{2} d x)

\int_{- r}^{r} r^{2} d x = 2 r^{3}

\int_{- r}^{r} x^{2} d x = \frac{2 r ^{3}}{3}

= π (2 r^{3} - \frac{2 r ^{3}}{3})