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Solving integrals/ y-y^2

Integral of y-y^2 dy

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫01(−y2+y) dy\int\limits_{0}^{1} \left(- y^{2} + y\right)\, dy 
Integral(y - y^2, (y, 0, 1))
Detail solution

  1. Integrate term-by-term:

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫(−y2) dy=−∫y2 dy\int \left(- y^{2}\right)\, dy = - \int y^{2}\, dy

      1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫y2 dy=y33\int y^{2}\, dy = \frac{y^{3}}{3}

      So, the result is: −y33- \frac{y^{3}}{3}

    1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫y dy=y22\int y\, dy = \frac{y^{2}}{2}

    The result is: −y33+y22- \frac{y^{3}}{3} + \frac{y^{2}}{2}

  2. Now simplify:

    y2(3−2y)6\frac{y^{2} \left(3 - 2 y\right)}{6}

  3. Add the constant of integration:

    y2(3−2y)6+constant\frac{y^{2} \left(3 - 2 y\right)}{6}+ \mathrm{constant}

The answer is:

y2(3−2y)6+constant\frac{y^{2} \left(3 - 2 y\right)}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                         
 |                    2    3
 | /     2\          y    y 
 | \y - y / dy = C + -- - --
 |                   2    3 
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∫(−y2+y) dy=C−y33+y22\int \left(- y^{2} + y\right)\, dy = C - \frac{y^{3}}{3} + \frac{y^{2}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.000.50
Plot the graph f(x)
The answer [src] 
1/6
16\frac{1}{6} 
=
= 
1/6
16\frac{1}{6} 
1/6
Numerical answer [src] 
0.166666666666667
0.166666666666667

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

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