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Solving integrals/ sqrt(5-y^2)

Integral of sqrt(5-y^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1               
  /               
 |                
 |     ________   
 |    /      2    
 |  \/  5 - y   dy
 |                
/                 
0
$$\int\limits_{0}^{1} \sqrt{5 - y^{2}}\, dy$$ 
Integral(sqrt(5 - y^2), (y, 0, 1))
Detail solution

    TrigSubstitutionRule(theta=_theta, func=sqrt(5)*sin(_theta), rewritten=5*cos(_theta)**2, substep=ConstantTimesRule(constant=5, 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=5*cos(_theta)**2, symbol=_theta), restriction=(y < sqrt(5)) & (y > -sqrt(5)), context=sqrt(5 - y**2), symbol=y)

  1. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                                                                                       
 |                      //      /    ___\                                                \
 |    ________          ||      |y*\/ 5 |        ________                                |
 |   /      2           ||5*asin|-------|       /      2                                 |
 | \/  5 - y   dy = C + |<      \   5   /   y*\/  5 - y           /       ___        ___\|
 |                      ||--------------- + -------------  for And\y > -\/ 5 , y < \/ 5 /|
/                       ||       2                2                                      |
                        \\                                                               /
$$\int \sqrt{5 - y^{2}}\, dy = C + \begin{cases} \frac{y \sqrt{5 - y^{2}}}{2} + \frac{5 \operatorname{asin}{\left(\frac{\sqrt{5} y}{5} \right)}}{2} & \text{for}\: y > - \sqrt{5} \wedge y < \sqrt{5} \end{cases}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
          /  ___\
          |\/ 5 |
    5*asin|-----|
          \  5  /
1 + -------------
          2
$$1 + \frac{5 \operatorname{asin}{\left(\frac{\sqrt{5}}{5} \right)}}{2}$$ 
=
= 
          /  ___\
          |\/ 5 |
    5*asin|-----|
          \  5  /
1 + -------------
          2
$$1 + \frac{5 \operatorname{asin}{\left(\frac{\sqrt{5}}{5} \right)}}{2}$$ 
1 + 5*asin(sqrt(5)/5)/2
Numerical answer [src] 
2.15911902250202
2.15911902250202

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

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