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Solving integrals/ sqrt(1-r^2)*r

Integral of sqrt(1-r^2)*r dr

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |     ________     
 |    /      2      
 |  \/  1 - r  *r dr
 |                  
/                   
0

$$\int\limits_{0}^{1} r \sqrt{1 - r^{2}}\, dr$$

Integral(sqrt(1 - r^2)*r, (r, 0, 1))

Detail solution

Let $u = 1 - r^{2}$ .

Then let $du = - 2 r dr$ and substitute $- \frac{du}{2}$ :

$\int \left(- \frac{\sqrt{u}}{2}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = - \frac{\int \sqrt{u}\, du}{2}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  So, the result is: $- \frac{u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

$- \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}$
Add the constant of integration:

$- \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                                3/2
 |    ________            /     2\   
 |   /      2             \1 - r /   
 | \/  1 - r  *r dr = C - -----------
 |                             3     
/

$$\int r \sqrt{1 - r^{2}}\, dr = C - \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/3

$$\frac{1}{3}$$

1/3

$$\frac{1}{3}$$

1/3

Numerical answer [src]

0.333333333333333

0.333333333333333

The graph

Use the examples entering the upper and lower limits of integration.

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Integral of sqrt(1-r^2)*r dr

Limits of integration:

The graph:

Piecewise:

The solution