What is the integral from r-h to r of pi(r^2-x^2) ?

The integral from r-h to r of pi(r^2-x^2) is pi(hr^2-(3hr^2-3h^2r+h^3)/3)

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

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

π (h r^{2} - \frac{3 h r ^{2} - 3 h ^{2} r + h ^{3}}{3})

Solution steps

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

Take the constant out:

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

= π \cdot \int_{r - h}^{r} r^{2} - x^{2} d x

Apply the Sum Rule:

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

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

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

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

= π (h r^{2} - \frac{3 h r ^{2} - 3 h ^{2} r + h ^{3}}{3})

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What is the integral from r-h to r of pi(r^2-x^2) ?
The integral from r-h to r of pi(r^2-x^2) is pi(hr^2-(3hr^2-3h^2r+h^3)/3)