The image contains three definite double integrals to be evaluated. Let's solve each one step by step:

Problem 1

$\int_{-3}^3 \int_0^{\pi/4} \left(y + y^3 \cos x\right) \, dx \, dy$

Problem 2

$\int_0^1 \int_0^3 e^{x + y} \, dx \, dy$

Problem 3

$\int_0^1 \int_y^1 e^{x/y} \, dx \, dy$

I'll compute each integral systematically. sympy as sp

Define symbols

x, y = sp.symbols('x y')

Problem 1: ∫∫ (y + y^3 * cos(x)) dxdy

integrand1 = y + y**3 * sp.cos(x) inner_integral1 = sp.integrate(integrand1, (x, 0, sp.pi/4)) outer_integral1 = sp.integrate(inner_integral1, (y, -3, 3))

Problem 2: ∫∫ e^(x+y) dxdy

integrand2 = sp.exp(x + y) inner_integral2 = sp.integrate(integrand2, (x, 0, 3)) outer_integral2 = sp.integrate(inner_integral2, (y, 0, 1))

Problem 3: ∫∫ e^(x/y) dxdy

integrand3 = sp.exp(x / y) inner_integral3 = sp.integrate(integrand3, (x, y, 1)) outer_integral3 = sp.integrate(inner_integral3, (y, 0, 1))

outer_integral1, outer_integral2, outer_integral3Here are the results of the three integrals:

1. Integral 1: $\int_{-3}^3 \int_0^{\pi/4} \left(y + y^3 \cos x\right) \, dx \, dy = 0$ This is because the function $y + y^3 \cos x$ is odd in $y$ over the symmetric limits $[-3, 3]$.

2. Integral 2: $\int_0^1 \int_0^3 e^{x+y} \, dx \, dy = -e^3 + 1 + e (e^3 - 1)$

3. Integral 3: The expression involves a divergence, as the result includes an improper integral component $\infty$, suggesting the integral does not converge.

Would you like more details or have questions about any specific part?

Further Questions

1. Why does symmetry in Problem 1 lead to a zero result?
2. How is the substitution in the exponential term handled for Problem 2?
3. Why does Problem 3 diverge at $y \to 0$?
4. Can numerical approximations be useful for divergent integrals like Problem 3?
5. How can symmetry properties simplify definite integrals?

Tip

Always check the behavior of the integrand near the integration boundaries to ensure convergence.