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sqrt(21^2-x^2)
Solving integrals/ sqrt(21^2-x^2)

Integral of sqrt(21^2-x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 21                 
  /                 
 |                  
 |     __________   
 |    /        2    
 |  \/  441 - x   dx
 |                  
/                   
0
$$\int\limits_{0}^{21} \sqrt{441 - x^{2}}\, dx$$ 
Integral(sqrt(441 - x^2), (x, 0, 21))
Detail solution

    TrigSubstitutionRule(theta=_theta, func=21*sin(_theta), rewritten=441*cos(_theta)**2, substep=ConstantTimesRule(constant=441, 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=441*cos(_theta)**2, symbol=_theta), restriction=(x > -21) & (x < 21), context=sqrt(441 - x**2), symbol=x)

  1. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                                                                                  
 |                                                                                   
 |    __________          //        /x \        __________                          \
 |   /        2           ||441*asin|--|       /        2                           |
 | \/  441 - x   dx = C + |<        \21/   x*\/  441 - x                            |
 |                        ||------------ + ---------------  for And(x > -21, x < 21)|
/                         \\     2                2                                 /
$$\int \sqrt{441 - x^{2}}\, dx = C + \begin{cases} \frac{x \sqrt{441 - x^{2}}}{2} + \frac{441 \operatorname{asin}{\left(\frac{x}{21} \right)}}{2} & \text{for}\: x > -21 \wedge x < 21 \end{cases}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
441*pi
------
  4
$$\frac{441 \pi}{4}$$ 
=
= 
441*pi
------
  4
$$\frac{441 \pi}{4}$$ 
441*pi/4
Numerical answer [src] 
346.360590058275
346.360590058275
The graph
Integral of sqrt(21^2-x^2) dx

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

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