To solve these integrals using Fubini's theorem, we decompose the double integrals into iterated integrals and evaluate them. Here's a quick explanation of Fubini's theorem before solving:

Fubini's Theorem

Fubini's theorem states that if a function $f(x, y)$ is continuous over a rectangular region $D$, the double integral can be computed as iterated integrals:

$\iint_D f(x, y) \, dx\, dy = \int_a^b \left( \int_c^d f(x, y) \, dy \right) dx = \int_c^d \left( \int_a^b f(x, y) \, dx \right) dy$

For non-rectangular regions, we use bounds determined by the region $D$.

1. Problem 1

$\iint_D \frac{xy}{1+x^2+y^2} \, dx\,dy, \quad D = \{(x, y) \in \mathbb{R}^2 \,|\, 0 \leq x \leq 1, 0 \leq y \leq 1 \leq x^2 + y^2 \}.$

To solve:

1. Bounds: $x \in [0, 1]$, ( y \in [0, \sqrt{.

Then explain Use theintegartimport sympy as sp

Variables

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

First integral

integrand1 = (x * y) / (1 + x2 + y2) region1_x = (0, 1) region1_y = (0, sp.sqrt(1 - x**2))

Solve the integral

inner_integral1 = sp.integrate(integrand1, (y, *region1_y)) result1 = sp.integrate(inner_integral1, (x, *region1_x))

Second integral

integrand2 = x * sp.cos(y) region2_y = (0, x**2) region2_x = (0, 2)

inner_integral2 = sp.integrate(integrand2, (y, *region2_y)) result2 = sp.integrate(inner_integral2, (x, *region2_x))

Third integral

integrand3 = x * y region3_y = (0, 2 - x) region3_x = (0, 2)

inner_integral3 = sp.integrate(integrand3, (y, *region3_y)) result3 = sp.integrate(inner_integral3, (x, *region3_x))

result1, result2, result3(1/4 - log(2)/4, 1/2 - cos(4)/2, 2/3)(1/4 - log(2)/4, 1/2 - cos(4)/2, 2/3)successHere are the solutions to the integrals using Fubini's theorem:

1. Integral 1: $\iint_D \frac{xy}{1+x^2+y^2} \, dx\, dy = \frac{1}{4} - \frac{\ln(2)}{4}.$

2. Integral 2: $\iint_D x \cos(y) \, dx\, dy = \frac{1}{2} - \frac{\cos(4)}{2}.$

3. Integral 3: $\iint_D x y \, dx\, dy = \frac{2}{3}.$

If you'd like detailed step-by-step solutions for any of these integrals, let me know!

Related Questions:

1. How do you determine the bounds of integration from the region $D$?
2. Can Fubini's theorem be applied to non-rectangular regions?
3. What happens if the integrand isn't continuous over $D$?
4. How would polar coordinates simplify the first integral?
5. How do you interpret the results of these integrals geometrically?

Tip: Always visualize the integration domain $D$ to confirm the bounds are correct before solving!