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Integral of (sqrt(8-2x^2)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  2                 
  /                 
 |                  
 |     __________   
 |    /        2    
 |  \/  8 - 2*x   dx
 |                  
/                   
-2                  
$$\int\limits_{-2}^{2} \sqrt{8 - 2 x^{2}}\, dx$$
Integral(sqrt(8 - 2*x^2), (x, -2, 2))
Detail solution
  1. Rewrite the integrand:

  2. 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=(x > -2) & (x < 2), context=sqrt(4 - x**2), symbol=x)

    So, the result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                 
 |                                                                                  
 |    __________                //                 ________                        \
 |   /        2             ___ ||                /      2                         |
 | \/  8 - 2*x   dx = C + \/ 2 *|<      /x\   x*\/  4 - x                          |
 |                              ||2*asin|-| + -------------  for And(x > -2, x < 2)|
/                               \\      \2/         2                              /
$$\int \sqrt{8 - 2 x^{2}}\, dx = C + \sqrt{2} \left(\begin{cases} \frac{x \sqrt{4 - x^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{x}{2} \right)} & \text{for}\: x > -2 \wedge x < 2 \end{cases}\right)$$
The graph
The answer [src]
       ___
2*pi*\/ 2 
$$2 \sqrt{2} \pi$$
=
=
       ___
2*pi*\/ 2 
$$2 \sqrt{2} \pi$$
2*pi*sqrt(2)
Numerical answer [src]
8.88576587631673
8.88576587631673

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