1. The problem asks for the general algorithm or steps to solve double integrals of the form $$\iint_R f(x,y) \, dx \, dy$$ over various regions $R$. 2. The general steps to solve such integrals are: 1. **Understand the region $R$:** Identify the limits of integration for $x$ and $y$. This can be rectangular (constant limits) or more complex (curves or inequalities). 2. **Set up the integral:** Write the double integral as an iterated integral, choosing the order of integration (either $dx \, dy$ or $dy \, dx$) based on the region's description. 3. **Integrate the inner integral:** Treat the outer variable as constant and integrate with respect to the inner variable. 4. **Integrate the outer integral:** Integrate the resulting expression with respect to the outer variable. 5. **Simplify the result:** Perform algebraic simplifications to get the final answer. 3. Important rules: - If the region $R$ is rectangular, limits are constants. - If $R$ is bounded by curves, express limits as functions. - Sometimes changing the order of integration simplifies the problem. - For complicated regions, consider coordinate transformations (polar, cylindrical, spherical). 4. Now, solve the first problem: **Problem 1:** Calculate $$\iint_R (x - 3y) \, dx \, dy$$ where $$R = [0,2] \times [1,2]$$. 5. Since $R$ is rectangular, limits are: $$0 \leq x \leq 2, \quad 1 \leq y \leq 2$$ 6. Set up the integral: $$\int_{y=1}^2 \int_{x=0}^2 (x - 3y) \, dx \, dy$$ 7. Integrate inner integral with respect to $x$: $$\int_0^2 (x - 3y) \, dx = \left[ \frac{x^2}{2} - 3yx \right]_0^2 = \frac{2^2}{2} - 3y \cdot 2 = 2 - 6y$$ 8. Now integrate outer integral with respect to $y$: $$\int_1^2 (2 - 6y) \, dy = \left[ 2y - 3y^2 \right]_1^2 = (4 - 12) - (2 - 3) = (-8) - (-1) = -7$$ 9. Final answer: $$\boxed{-7}$$