Mister Exam

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Similar expressions
a(x^2-y^2)
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Integral of -a(x^2-y^2) dy

The solution

You have entered [src]

  1                
  /                
 |                 
 |     / 2    2\   
 |  -a*\x  - y / dy
 |                 
/                  
0

$$\int\limits_{0}^{1} - a \left(x^{2} - y^{2}\right)\, dy$$

Integral((-a)*(x^2 - y^2), (y, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int - a \left(x^{2} - y^{2}\right)\, dy = - a \int \left(x^{2} - y^{2}\right)\, dy$
1. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int x^{2}\, dy = x^{2} y$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- y^{2}\right)\, dy = - \int y^{2}\, dy$
    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: $- \frac{y^{3}}{3}$
  The result is: $x^{2} y - \frac{y^{3}}{3}$
So, the result is: $- a \left(x^{2} y - \frac{y^{3}}{3}\right)$
Now simplify:

$\frac{a y \left(- 3 x^{2} + y^{2}\right)}{3}$
Add the constant of integration:

$\frac{a y \left(- 3 x^{2} + y^{2}\right)}{3}+ \mathrm{constant}$

The answer is:

$\frac{a y \left(- 3 x^{2} + y^{2}\right)}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                         /   3       \
 |    / 2    2\            |  y       2|
 | -a*\x  - y / dy = C - a*|- -- + y*x |
 |                         \  3        /
/

$$\int - a \left(x^{2} - y^{2}\right)\, dy = C - a \left(x^{2} y - \frac{y^{3}}{3}\right)$$

Simplify

The answer [src]

a      2
- - a*x 
3

$$- a x^{2} + \frac{a}{3}$$

a      2
- - a*x 
3

$$- a x^{2} + \frac{a}{3}$$

a/3 - a*x^2

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

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Identical expressions

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Integral of -a(x^2-y^2) dy

Limits of integration:

The graph:

Piecewise:

The solution