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Identical expressions
sqrt(four -y^ two)-(y^ two)/ three
square root of (4 minus y squared ) minus (y squared ) divide by 3
square root of (four minus y to the power of two) minus (y to the power of two) divide by three
√(4-y^2)-(y^2)/3
sqrt(4-y2)-(y2)/3
sqrt4-y2-y2/3
sqrt(4-y²)-(y²)/3
sqrt(4-y to the power of 2)-(y to the power of 2)/3
sqrt4-y^2-y^2/3
sqrt(4-y^2)-(y^2) divide by 3
Similar expressions
sqrt(4+y^2)-(y^2)/3
sqrt(4-y^2)+(y^2)/3

Integral of sqrt(4-y^2)-(y^2)/3 dx

The solution

You have entered [src]

    ___                     
  \/ 3                      
    /                       
   |                        
   |   /   ________    2\   
   |   |  /      2    y |   
   |   |\/  4 - y   - --| dy
   |   \              3 /   
   |                        
  /                         
   ___                      
-\/ 3

$$\int\limits_{- \sqrt{3}}^{\sqrt{3}} \left(- \frac{y^{2}}{3} + \sqrt{4 - y^{2}}\right)\, dy$$

Integral(sqrt(4 - y^2) - y^2/3, (y, -sqrt(3), sqrt(3)))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{y^{2}}{3}\right)\, dy = - \frac{\int y^{2}\, dy}{3}$
  1. The integral of $y^{n}$ is $\frac{y^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int y^{2}\, dy = \frac{y^{3}}{3}$
  So, the result is: $- \frac{y^{3}}{9}$
The result is: $- \frac{y^{3}}{9} + \begin{cases} \frac{y \sqrt{4 - y^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{y}{2} \right)} & \text{for}\: y > -2 \wedge y < 2 \end{cases}$
Now simplify:

$\begin{cases} - \frac{y^{3}}{9} + \frac{y \sqrt{4 - y^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{y}{2} \right)} & \text{for}\: y > -2 \wedge y < 2 \end{cases}$
Add the constant of integration:

$\begin{cases} - \frac{y^{3}}{9} + \frac{y \sqrt{4 - y^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{y}{2} \right)} & \text{for}\: y > -2 \wedge y < 2 \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} - \frac{y^{3}}{9} + \frac{y \sqrt{4 - y^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{y}{2} \right)} & \text{for}\: y > -2 \wedge y < 2 \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                     
 |                                                                                      
 | /   ________    2\           3   //                 ________                        \
 | |  /      2    y |          y    ||                /      2                         |
 | |\/  4 - y   - --| dy = C - -- + |<      /y\   y*\/  4 - y                          |
 | \              3 /          9    ||2*asin|-| + -------------  for And(y > -2, y < 2)|
 |                                  \\      \2/         2                              /
/

$$\int \left(- \frac{y^{2}}{3} + \sqrt{4 - y^{2}}\right)\, dy = C - \frac{y^{3}}{9} + \begin{cases} \frac{y \sqrt{4 - y^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{y}{2} \right)} & \text{for}\: y > -2 \wedge y < 2 \end{cases}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ___       
\/ 3    4*pi
----- + ----
  3      3

$$\frac{\sqrt{3}}{3} + \frac{4 \pi}{3}$$

  ___       
\/ 3    4*pi
----- + ----
  3      3

$$\frac{\sqrt{3}}{3} + \frac{4 \pi}{3}$$

sqrt(3)/3 + 4*pi/3

Numerical answer [src]

4.76614047397602

4.76614047397602

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

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Integral of sqrt(4-y^2)-(y^2)/3 dx

Limits of integration:

The graph:

Piecewise:

The solution