Solve ∫ (from 0 to r) of 1/2(sqrt{r^2-x^2})^2 wrt x

$$\int _ { 0 } ^ { r } \frac { 1 } { 2 } ( \sqrt { r ^ { 2 } - x ^ { 2 } } ) ^ { 2 } d x$$

Evaluate

$$\frac{r^{3}}{3}$$

Solution Steps

Calculate \sqrt{r^{2}-x^{2}} to the power of 2 and get r^{2}-x^{2}.

$$\int _{0}^{r}\frac{1}{2}\left(r^{2}-x^{2}\right)\mathrm{d}x$$

Use the distributive property to multiply \frac{1}{2} by r^{2}-x^{2}.

$$\int _{0}^{r}\frac{1}{2}r^{2}-\frac{1}{2}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{r^{2}-x^{2}}{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{r^{2}}{2}\mathrm{d}x+\int -\frac{x^{2}}{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{\int r^{2}\mathrm{d}x-\int x^{2}\mathrm{d}x}{2}$$

Find the integral of r^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{r^{2}x-\int x^{2}\mathrm{d}x}{2}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -\frac{1}{2} times \frac{x^{3}}{3}.

$$\frac{r^{2}x}{2}-\frac{x^{3}}{6}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}r^{2}r-\frac{r^{3}}{6}-\left(\frac{1}{2}r^{2}\times 0-\frac{0^{3}}{6}\right)$$

Simplify.

$$\frac{r^{3}}{3}$$

Differentiate w.r.t. r

$$r^{2}$$