2.9: Chapter 2 Review Exercises

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Apr 3, 2025
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- 2.8E: Exercises
- 3: Vector Calculus

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Exercises 1 - 4

True or False? Justify your answer with a proof or a counterexample.

1. $\displaystyle ∫_a^b∫_c^d f(x,y) \, dy \, dx = ∫_c^d∫_a^b f(x,y) \, dy \, dx$

2. Fubini’s theorem can be extended to three dimensions, as long as $f$ is continuous in all variables.

Answer: True

3. The integral $\displaystyle ∫_0^{2π}∫_0^1∫_r^1 \,dz \, dr \, dθ$ represents the volume of a right cone.

4. The Jacobian of the transformation for $x=u^2−2v, \, y=3v−2uv$ is given by $−4u^2+6u+4v.$

Answer: False

Exercises 5 - 13

Evaluate the following integrals.

5. $\displaystyle \iint_R (5x^3y^2−y^2) \, dA,$ where $R=\big\{(x,y) \,|\, 0≤x≤2,\, 1≤y≤4\big\}$

6. $\displaystyle \iint_D \dfrac{y}{3x^2+1} \, dA,$ where $D=\big\{(x,y) \,|\, 0≤x≤1, \, −x≤y≤x\big\}$

Answer: $0$

7. $\displaystyle \iint_D \sin(x^2+y^2) \, dA$ where $D$ is a disk of radius $2$ centered at the origin.

8. $\displaystyle \int_0^1\int_y^1 xye^{x^2}\,dx \, dy$

Answer: $\frac{1}{4}$

9. $\displaystyle \int_{−1}^1\int_0^z\int_0^{x−z} 6 \, dy \, dx\, dz$

10. $\displaystyle \iiint_R 3y \, dV,$ where $R=\big\{(x,y,z) \,|\, 0≤x≤1, \, 0≤y≤x, \, 0≤z≤9−y^2\big\}$

Answer: $1.475$

11. $\displaystyle \int_0^2\int_0^{2π}\int_r^1 r \, dz \, dθ \, dr$

12. $\displaystyle \int_0^{2π}\int_0^{π/2}\int_1^3 ρ^2\sin(φ) \, dρ \, dφ \, dθ$

Answer: $\frac{52\pi}{3}$

13. $\displaystyle \int_0^1\int_{−\sqrt{1−x^2}}^{\sqrt{1−x^2}}\int_{−\sqrt{1−x^2−y^2}}^{\sqrt{1−x^2−y^2}} \, dz \, dy \, dx$

Exercises 14 - 17

For the following problems, find the specified area or volume.

14. The area of region enclosed by one petal of $r=\cos(4θ).$

Answer: $\frac{\pi}{16} \text{ units}^3$

15. The volume of the solid that lies between the paraboloid $z=2x^2+2y^2$ and the plane $z=8.$

16. The volume of the solid bounded by the cylinder $x^2+y^2=16$ and from $z=1$ to $z+x=2.$

Answer: $93.291 \text{ units}^3$

17. The volume of the intersection between two spheres of radius $1,$ the top whose center is $(0,\,0,\,0.25)$ and the bottom, which is centered at $(0,\,0,\,0).$

Exercises 18 - 21

For the following problems, find the center of mass of the region.

18. $ρ(x,y)=xy$ on the circle with radius $1$ in the first quadrant only.

Answer: $\left( \frac{8}{15}, \, \frac{8}{15} \right)$

19. $ρ(x,y)=(y+1)\sqrt{x}$ in the region bounded by $y=e^x, \, y=0,$ and $x=1.$

20. $ρ(x,y,z)=z$ on the inverted cone with radius $2$ and height $2.$

Answer: $\left( 0, \, 0, \, \frac{8}{5} \right)$

21. The volume an ice cream cone that is given by the solid above $z=\sqrt{x^2+y^2}$ and below $z^2+x^2+y^2=z.$

Exercises 22 - 23

The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a right circular cone of height 1100 ft and radius 6000 ft.

22. If the compacted trash used to build Mount Holly on average has a density $400\text{ lb/ft}^3,$ find the amount of work required to build the mountain.

Answer: $1.452\pi \times 10^{15}$ ft-lb

23. In reality, it is very likely that the trash at the bottom of Mount Holly has become more compacted with all the weight of the above trash. Consider a density function with respect to height: the density at the top of the mountain is still density $400\text{ lb/ft}^3,$ and the density increases. Every 100 feet deeper, the density doubles. What is the total weight of Mount Holly?

Exercises 24 - 25

The following problems consider the temperature and density of Earth’s layers.

24. The temperature of Earth’s layers is exhibited in the table below. Use your calculator to fit a polynomial of degree 3 to the temperature along the radius of the Earth. Then find the average temperature of Earth. (Hint: begin at 0 in the inner core and increase outward toward the surface)

Table $\PageIndex{1}$ : The temperature of Earth’s layers is exhibited in the table.
Layer	Depth from center (km)	Temperature °C
Rocky Crust	0 to 40	0
Upper Mantle	40 to 150	870
Mantle	400 to 650	870
Inner Mantel	650 to 2700	870
Molten Outer Core	2890 to 5150	4300
Inner Core	5150 to 6378	7200

Answer: $y=−1.238×10^{−7}x^3+0.001196x^2−3.666x+7208$ ;
The average temperature is approximately 2800 °C.

25. The density of Earth’s layers is displayed in the table below. Using your calculator or a computer program, find the best-fit quadratic equation to the density. Using this equation, find the total mass of Earth.

Table $\PageIndex{2}$ : The table includes density of Earth’s layers.
Layer	Depth from center (km)	Density ( $\text{g/cm}^3$ )
Inner Core	0	12.95
Outer Core	1228	11.05
Mantle	3488	5.00
Upper Mantle	6338	3.90
Crust	6378	2.55

Exercises 26 - 27

The following problems concern the Theorem of Pappus (see Moments and Centers of Mass for a refresher), a method for calculating volume using centroids. Assuming a region $R$ , when you revolve around the $x$ -axis the volume is given by $V_x=2πA\overline{y},$ and when you revolve around the $y$ -axis the volume is given by $V_y=2πA\overline{x},$ where $A$ is the area of $R$ . Consider the region bounded by $x^2+y^2=1$ and above $y=x+1$ .

26. Find the volume when you revolve the region around the $x$ -axis.

Answer: $\frac{\pi}{3} \text{ units}^3$

27. Find the volume when you revolve the region around the $y$ -axis.

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