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

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

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The solution

You have entered [src] 
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∫01r1−r2 dr\int\limits_{0}^{1} r \sqrt{1 - r^{2}}\, dr 
Integral(sqrt(1 - r^2)*r, (r, 0, 1))
Detail solution

  1. Let u=1−r2u = 1 - r^{2}.

    Then let du=−2rdrdu = - 2 r dr and substitute −du2- \frac{du}{2}:

    ∫(−u2) du\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:

      ∫u du=−∫u du2\int \sqrt{u}\, du = - \frac{\int \sqrt{u}\, du}{2}

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u du=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

      So, the result is: −u323- \frac{u^{\frac{3}{2}}}{3}

    Now substitute uu back in:

    −(1−r2)323- \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}

  2. Add the constant of integration:

    −(1−r2)323+constant- \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer is:

−(1−r2)323+constant- \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫r1−r2 dr=C−(1−r2)323\int r \sqrt{1 - r^{2}}\, dr = C - \frac{\left(1 - r^{2}\right)^{\frac{3}{2}}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901.0-1.0
Plot the graph f(x)
The answer [src] 
1/3
13\frac{1}{3} 
=
= 
1/3
13\frac{1}{3} 
1/3
Numerical answer [src] 
0.333333333333333
0.333333333333333
The graph
Integral of sqrt(1-r^2)*r dr

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

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