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Solving integrals/ 2*sqrt(4-y^2)

Integral of 2*sqrt(4-y^2) dy

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  2                 
  /                 
 |                  
 |       ________   
 |      /      2    
 |  2*\/  4 - y   dy
 |                  
/                   
0
$$\int\limits_{0}^{2} 2 \sqrt{4 - y^{2}}\, dy$$ 
Integral(2*sqrt(4 - y^2), (y, 0, 2))
Detail solution

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

      TrigSubstitutionRule(theta=_theta, func=2*sin(_theta), rewritten=4*cos(_theta)**2, substep=ConstantTimesRule(constant=4, other=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), context=4*cos(_theta)**2, symbol=_theta), restriction=(y > -2) & (y < 2), context=sqrt(4 - y**2), symbol=y)

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                                                                             
 |                                                                              
 |      ________            //                 ________                        \
 |     /      2             ||                /      2                         |
 | 2*\/  4 - y   dy = C + 2*|<      /y\   y*\/  4 - y                          |
 |                          ||2*asin|-| + -------------  for And(y > -2, y < 2)|
/                           \\      \2/         2                              /
$$\int 2 \sqrt{4 - y^{2}}\, dy = C + 2 \left(\begin{cases} \frac{y \sqrt{4 - y^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{y}{2} \right)} & \text{for}\: y > -2 \wedge y < 2 \end{cases}\right)$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
2*pi
$$2 \pi$$ 
=
= 
2*pi
$$2 \pi$$ 
2*pi
Numerical answer [src] 
6.28318530717959
6.28318530717959

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

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