Partial Derivatives

#Example 4: Section 14.1, Thomas 15th edition.
from sympy import symbols, integrate

# Define the variables
x, y = symbols('x y')

# Define the function to integrate
f = 16 - x**2 - y**2

# Define the bounds
x_lower = y**2 / 4
x_upper = (y + 2)/4
y_lower = 0
y_upper = 2

# Compute the double integral
res = integrate(f, (x, x_lower, x_upper))
result = integrate(res, (y, y_lower, y_upper))

# Display the result
display(res)
print('Now integrating over y')
display(result)

\(\displaystyle \frac{y^{6}}{192} - \frac{y^{2} \left(16 - y^{2}\right)}{4} + \left(16 - y^{2}\right) \left(\frac{y}{4} + \frac{1}{2}\right) - \frac{\left(\frac{y}{4} + \frac{1}{2}\right)^{3}}{3}\)

Now integrating over y

\(\displaystyle \frac{20803}{1680}\)