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Practice

Working with scipy.integrate


The scipy.integrate module provides tools for numerical integration and for solving ordinary differential equations (ODEs).

It’s especially useful when:

  • You need to compute definite integrals
  • You want to integrate functions without closed-form solutions
  • You’re solving systems of differential equations

Example 1: Definite Integral with quad​

Use integrate.quad() to compute the definite integral of a function within a given range.

Definite Integral of sin(x) from 0 to π
# Integrand
f = lambda x: np.sin(x)

# Integrate from 0 to π
val, err = integrate.quad(f, 0, np.pi)

print("Integral value:", val)
print("Estimated error:", err)

Explanation:​

  • quad() returns both the integral value and an estimated numerical error.
  • Best suited for smooth, continuous functions.

Example 2: Solving an ODE with solve_ivp​

Use integrate.solve_ivp() to solve initial value problems (IVPs) for differential equations.

Solve y' = -2y, y(0) = 1
# Derivative function
def dydt(t, y):
    return -2 * y

# Time span and evaluation points
t_span = (0, 3)
t_eval = np.linspace(*t_span, 100)

# Solve the ODE
sol = integrate.solve_ivp(dydt, t_span, y0=[1.0], t_eval=t_eval)

print("First 5 y values:", sol.y[0][:5])

Explanation:​

  • solve_ivp() numerically integrates an ODE over a time interval.
  • t_span defines the range of integration.
  • y0 provides the initial conditions for the system.

Key Takeaways​

  • quad() — integrates single-variable functions over a defined range.
  • solve_ivp() — solves initial value problems for ordinary differential equations.
  • Use NumPy arrays when defining functions to ensure full compatibility with SciPy’s integration tools.

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