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Solving integrals/ sqrt(4-y)

Integral of sqrt(4-y) dy

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src] 
  4             
  /             
 |              
 |    _______   
 |  \/ 4 - y  dy
 |              
/               
0
∫044−y dy\int\limits_{0}^{4} \sqrt{4 - y}\, dy 
Integral(sqrt(4 - y), (y, 0, 4))
Detail solution

  1. Let u=4−yu = 4 - y.

    Then let du=−dydu = - dy and substitute −du- du:

    ∫(−u) du\int \left(- \sqrt{u}\right)\, du

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

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

      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: −2u323- \frac{2 u^{\frac{3}{2}}}{3}

    Now substitute uu back in:

    −2(4−y)323- \frac{2 \left(4 - y\right)^{\frac{3}{2}}}{3}

  2. Add the constant of integration:

    −2(4−y)323+constant- \frac{2 \left(4 - y\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer is:

−2(4−y)323+constant- \frac{2 \left(4 - y\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
 |                             3/2
 |   _______          2*(4 - y)   
 | \/ 4 - y  dy = C - ------------
 |                         3      
/
∫4−y dy=C−2(4−y)323\int \sqrt{4 - y}\, dy = C - \frac{2 \left(4 - y\right)^{\frac{3}{2}}}{3}
Simplify
The graph
0.04.00.51.01.52.02.53.03.5-1010
Plot the graph f(x)
The answer [src] 
16/3
163\frac{16}{3} 
=
= 
16/3
163\frac{16}{3} 
16/3
Numerical answer [src] 
5.33333333333333
5.33333333333333

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

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