1. **Problem statement:** Find the volume of the solid bounded by the surface $f(x,y)$ over the given region $R$. We will solve problems 3, 4, and 12 from Exercise 2.2. --- ### Problem 3 2. **Given:** $f(x,y) = 1 + 4xy$, region $R: 0 \leq x \leq 1$, $1 \leq y \leq 3$. 3. **Formula:** Volume $V = \iint_R f(x,y) \, dA = \int_{x=a}^{b} \int_{y=c}^{d} f(x,y) \, dy \, dx$. 4. **Calculate:** $$V = \int_0^1 \int_1^3 (1 + 4xy) \, dy \, dx$$ 5. **Inner integral:** $$\int_1^3 (1 + 4xy) \, dy = \int_1^3 1 \, dy + \int_1^3 4xy \, dy = (y)\big|_1^3 + 4x \frac{y^2}{2} \bigg|_1^3 = (3 - 1) + 4x \left(\frac{9}{2} - \frac{1}{2}\right) = 2 + 4x \cdot 4 = 2 + 16x$$ 6. **Outer integral:** $$\int_0^1 (2 + 16x) \, dx = 2x + 8x^2 \bigg|_0^1 = 2 + 8 = 10$$ 7. **Answer:** The volume is $\boxed{10}$. --- ### Problem 4 8. **Given:** $f(x,y) = xy - 3xy^2$, region $R: 0 \leq x \leq 2$, $1 \leq y \leq 2$. 9. **Calculate:** $$V = \int_0^2 \int_1^2 (xy - 3xy^2) \, dy \, dx = \int_0^2 x \int_1^2 (y - 3y^2) \, dy \, dx$$ 10. **Inner integral:** $$\int_1^2 (y - 3y^2) \, dy = \left(\frac{y^2}{2} - y^3\right) \bigg|_1^2 = \left(\frac{4}{2} - 8\right) - \left(\frac{1}{2} - 1\right) = (2 - 8) - (0.5 - 1) = -6 + 0.5 = -5.5$$ 11. **Outer integral:** $$\int_0^2 x (-5.5) \, dx = -5.5 \int_0^2 x \, dx = -5.5 \cdot \frac{x^2}{2} \bigg|_0^2 = -5.5 \cdot 2 = -11$$ 12. **Answer:** The volume is $\boxed{-11}$. --- ### Problem 12 13. **Given:** $f(x,y) = x \sin(xy)$, region $R: 0 \leq x \leq \pi$, $1 \leq y \leq 2$. 14. **Calculate:** $$V = \int_0^{\pi} \int_1^2 x \sin(xy) \, dy \, dx$$ 15. **Inner integral:** Let $x$ be constant, $$\int_1^2 \sin(xy) \, dy = \frac{-\cos(xy)}{x} \bigg|_1^2 = \frac{-\cos(2x) + \cos(x)}{x}$$ 16. Multiply by $x$: $$x \cdot \int_1^2 \sin(xy) \, dy = -\cos(2x) + \cos(x)$$ 17. **Outer integral:** $$\int_0^{\pi} [-\cos(2x) + \cos(x)] \, dx = \int_0^{\pi} -\cos(2x) \, dx + \int_0^{\pi} \cos(x) \, dx$$ 18. Evaluate each: $$\int_0^{\pi} -\cos(2x) \, dx = -\frac{\sin(2x)}{2} \bigg|_0^{\pi} = 0$$ $$\int_0^{\pi} \cos(x) \, dx = \sin(x) \bigg|_0^{\pi} = 0$$ 19. Sum: $$0 + 0 = 0$$ 20. **Answer:** The volume is $\boxed{0}$. --- **Summary:** - Problem 3 volume: $10$ - Problem 4 volume: $-11$ - Problem 12 volume: $0$