```python
from sympy import symbols, oo, integrate, exp, pi, sin, cos, sqrt
```

## Going to Khan Academy to review fundamentals

[https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/double-integrals-a/a/double-integrals](https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/double-integrals-a/a/double-integrals)


```python
x, y = symbols('x y', real=True)
```


```python
inner = integrate(x+sin(y)+1, (x, 0., 2.))
inner
```




$\displaystyle 2.0 \sin{\left(y \right)} + 4.0$




```python
outer = integrate(inner, (y, -pi, pi))
outer
```




$\displaystyle 8.0 \pi$




```python
inner = integrate(x+sin(y)+1, (y, -pi, pi))
inner
```




$\displaystyle 2 \pi x + 2 \pi$




```python
outer = integrate(inner, (x, 0., 2.))
outer
```




$\displaystyle 8.0 \pi$




```python
inner = integrate(cos(y)*x**2+1, (x, -1., 1.))
inner
```




$\displaystyle 0.666666666666667 \cos{\left(y \right)} + 2.0$




```python
outer = integrate(inner, (y, -pi, pi))
outer
```




$\displaystyle 4.0 \pi$



[https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/double-integrals-a/a/double-integrals-over-non-rectangular-regions](https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/double-integrals-a/a/double-integrals-over-non-rectangular-regions)


```python
inner = integrate(x*y**2, (y, 0, x))
inner
```




$\displaystyle \frac{x^{4}}{3}$




```python
outer = integrate(x**4/3., (x, 0., 2.))
outer
```




$\displaystyle 2.13333333333333$




```python
lower = -sqrt(1-x**2)
upper = sqrt(1-x**2)
integrate(3+y-x**2, (y, lower, upper))
```




$\displaystyle 2 \sqrt{1 - x^{2}} \left(3 - x^{2}\right)$




```python
upper = y+2
lower = y**2
inner = integrate(x+2*y, (x, lower, upper))
inner
```




$\displaystyle - \frac{y^{4}}{2} - 2 y^{3} + 2 y \left(y + 2\right) + \frac{\left(y + 2\right)^{2}}{2}$




```python
outer = integrate(inner, (y, 0., 2.))
outer
```




$\displaystyle 11.4666666666667$



[https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/double-integrals-a/a/double-integrals-in-polar-coordinates](https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/double-integrals-a/a/double-integrals-in-polar-coordinates)


```python
r, theta = symbols('r theta', real=True, positive=True)
```


```python
inner = integrate(r**3, (theta, 0, 2*pi))
inner
```




$\displaystyle 2 \pi r^{3}$




```python
outer = integrate(inner, (r, 0., 2.))
outer
```




$\displaystyle 8.0 \pi$




```python
inner = integrate(r**2*sin(theta), (r, 0., cos(2*theta)))
inner
```




$\displaystyle \frac{\sin{\left(\theta \right)} \cos^{3}{\left(2 \theta \right)}}{3}$




```python
outer = integrate(inner, (theta, 0., 2*pi))
outer
```




$\displaystyle 0$




```python

```

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notebooks/double_integrals_khan_academy.ipynb
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2021-07-12T15:49:00.000Z
2021-07-12T15:54:55.000Z
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akey7/basic-quantum-models
b2f7edd2fdd15b6f85ee3708f8c2dae24686fd07
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2. YES