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Identical expressions
(sqrt(eight - two x^2))
( square root of (8 minus 2x squared ))
( square root of (eight minus two x squared ))
(√(8-2x^2))
(sqrt(8-2x2))
sqrt8-2x2
(sqrt(8-2x²))
(sqrt(8-2x to the power of 2))
sqrt8-2x^2
(sqrt(8-2x^2))dx
Similar expressions
(sqrt(8+2x^2))

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

The solution

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  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

Rewrite the integrand:

$\sqrt{8 - 2 x^{2}} = \sqrt{2} \sqrt{4 - x^{2}}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \sqrt{2} \sqrt{4 - x^{2}}\, dx = \sqrt{2} \int \sqrt{4 - x^{2}}\, dx$
So, the result is: $\sqrt{2} \left(\begin{cases} \frac{x \sqrt{4 - x^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{x}{2} \right)} & \text{for}\: x > -2 \wedge x < 2 \end{cases}\right)$
Now simplify:

$\begin{cases} \frac{\sqrt{2} \left(x \sqrt{4 - x^{2}} + 4 \operatorname{asin}{\left(\frac{x}{2} \right)}\right)}{2} & \text{for}\: x > -2 \wedge x < 2 \end{cases}$
Add the constant of integration:

$\begin{cases} \frac{\sqrt{2} \left(x \sqrt{4 - x^{2}} + 4 \operatorname{asin}{\left(\frac{x}{2} \right)}\right)}{2} & \text{for}\: x > -2 \wedge x < 2 \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \frac{\sqrt{2} \left(x \sqrt{4 - x^{2}} + 4 \operatorname{asin}{\left(\frac{x}{2} \right)}\right)}{2} & \text{for}\: x > -2 \wedge x < 2 \end{cases}+ \mathrm{constant}$

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                              /

$$\int \sqrt{8 - 2 x^{2}}\, dx = C + \sqrt{2} \left(\begin{cases} \frac{x \sqrt{4 - x^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{x}{2} \right)} & \text{for}\: x > -2 \wedge x < 2 \end{cases}\right)$$

Simplify

The graph

Plot the graph f(x)

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.

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

Limits of integration:

The graph:

Piecewise:

The solution