Chapter 32 — Exercises

38 problems on double and triple integrals, reversing order, polar/cylindrical/spherical coordinates, mass, center of mass, moment of inertia, the Gaussian integral, and probability. ⭐ to ⭐⭐⭐⭐.

Work these with paper, pencil, and a clear sketch of every region. Multiple integration rewards the patient: draw the region first, decide on the order or coordinate system, and only then start computing. Answers to odd-numbered problems appear in appendices/answers-to-selected.md.

Difficulty key: ⭐ routine · ⭐⭐ standard · ⭐⭐⭐ challenging · ⭐⭐⭐⭐ synthesis.

Part A — Double Integrals over Rectangles (Fubini) ⭐

These rely only on §32.2: integrate the inner variable holding the other fixed, then the outer.

1. ⭐ Evaluate $\displaystyle\int_0^2\int_0^3 (x + y)\,dy\,dx$.

2. ⭐ Evaluate $\displaystyle\iint_R xy\,dA$ over $R = [0,2]\times[0,3]$.

3. ⭐ Evaluate $\displaystyle\iint_R (2x + 4y)\,dA$ over $R = [0,1]\times[0,2]$.

4. ⭐ Evaluate $\displaystyle\int_0^1\int_0^1 e^{x+y}\,dx\,dy$. (Hint: $e^{x+y} = e^x e^y$ — use the factoring shortcut.)

5. ⭐ Evaluate $\displaystyle\iint_R \cos x \,\sin y\,dA$ over $R = [0,\pi/2]\times[0,\pi/2]$ using the product shortcut.

6. ⭐ Confirm Fubini for $\displaystyle\iint_R x^2 y\,dA$ over $R = [0,1]\times[0,2]$ by computing in both orders and showing they agree.

7. ⭐ Evaluate $\displaystyle\int_1^2\int_1^2 \frac{1}{xy}\,dx\,dy$.

8. ⭐ Find the average value of $f(x,y) = x + y$ over $R = [0,2]\times[0,2]$ (use §32.4).

9. ⭐ Evaluate $\displaystyle\iint_R 1\,dA$ over $R = [-1,3]\times[2,5]$ and verify it equals the rectangle's area.

Part B — General Regions and Reversing Order ⭐⭐

Sketch each region (§32.3). For reversals, re-derive the new limits from the picture — never shuffle the old ones.

10. ⭐⭐ Evaluate $\displaystyle\int_0^1\int_0^x (x + y)\,dy\,dx$ over the triangle with vertices $(0,0),(1,0),(1,1)$.

11. ⭐⭐ Evaluate $\displaystyle\iint_R x\,dA$ where $R$ is bounded by $y = x^2$ and $y = x$ (these meet at $(0,0)$ and $(1,1)$).

12. ⭐⭐ Evaluate $\displaystyle\iint_R y\,dA$ over the region under $y = \sqrt{x}$, above $y = 0$, between $x = 0$ and $x = 4$.

13. ⭐⭐ Find the area enclosed between $y = x^2$ and $y = 2 - x^2$ using a double integral of $f \equiv 1$.

14. ⭐⭐ Set up both the Type I and Type II iterated integrals for $\iint_R f\,dA$ over the region bounded by $y = 0$, $x = 1$, and $y = x$. (Do not evaluate; describe the limits.)

15. ⭐⭐ Evaluate $\displaystyle\int_0^1\int_{y}^{1} e^{x^2}\,dx\,dy$ by reversing the order of integration.

16. ⭐⭐ Evaluate $\displaystyle\int_0^2\int_{x/2}^{1} \cos(y^2)\,dy\,dx$ by reversing the order.

17. ⭐⭐ Reverse the order of integration in $\displaystyle\int_0^1\int_{x^2}^{1} f(x,y)\,dy\,dx$ and write the result (limits only).

18. ⭐⭐ Evaluate $\displaystyle\iint_R (x + y)\,dA$ over the triangle with vertices $(0,0),(2,0),(0,2)$.

19. ⭐⭐ Find the volume under $z = 4 - x - y$ over the triangle $0 \le x \le 1$, $0 \le y \le 1 - x$.

20. ⭐⭐ Find the centroid $(\bar x, \bar y)$ of the region bounded by $y = x^2$ and $y = 4$ (uniform density). Use the symmetry to predict $\bar x$ before computing.

21. ⭐⭐ A flat plate occupies the triangle with vertices $(0,0),(2,0),(0,4)$ and has density $\rho(x,y) = x + y$. Find its mass.

22. ⭐⭐ Evaluate $\displaystyle\iint_R \frac{y}{1 + x^2}\,dA$ over $R = \{0 \le x \le 1,\ 0 \le y \le x\}$.

23. ⭐⭐ Reverse the order and evaluate $\displaystyle\int_0^4\int_{\sqrt{x}}^{2} \frac{1}{y^3 + 1}\,dy\,dx$.

Part C — Polar Coordinates, the Gaussian, and Triple Integrals ⭐⭐⭐

Remember the polar area element $dA = r\,dr\,d\theta$ — the $r$ is never optional (§32.5).

24. ⭐⭐⭐ Evaluate $\displaystyle\iint_D (x^2 + y^2)\,dA$ over the disk $D : x^2 + y^2 \le 4$ using polar coordinates.

25. ⭐⭐⭐ Find the area of one petal of the four-petal rose $r = \cos 2\theta$ (one petal spans $-\pi/4 \le \theta \le \pi/4$). Recall the polar-area formula $\frac12\int r^2\,d\theta$ from Chapter 26 as a check.

26. ⭐⭐⭐ Evaluate $\displaystyle\iint_D e^{-(x^2+y^2)}\,dA$ over the disk $x^2 + y^2 \le 4$, and compare to the whole-plane value $\pi$.

27. ⭐⭐⭐ Find the volume of the solid bounded above by the paraboloid $z = 9 - x^2 - y^2$ and below by the plane $z = 0$ (use polar).

28. ⭐⭐⭐ Evaluate $\displaystyle\int_0^a\int_0^{\sqrt{a^2 - x^2}} \sqrt{x^2 + y^2}\,dy\,dx$ by converting to polar (the region is a quarter-disk of radius $a$).

29. ⭐⭐⭐ Evaluate the triple integral $\displaystyle\iiint_E z\,dV$ where $E$ is the box $[0,1]\times[0,2]\times[0,3]$.

30. ⭐⭐⭐ Find the volume of the tetrahedron bounded by the coordinate planes and the plane $x + y + z = 1$ using a triple integral.

31. ⭐⭐⭐ Use cylindrical coordinates to find the volume of the solid inside the cylinder $x^2 + y^2 = 1$, between $z = 0$ and $z = 4 - x^2 - y^2$.

32. ⭐⭐⭐ Use spherical coordinates to find the volume of the region above the cone $\phi = \pi/4$ and inside the sphere $\rho = 2$.

33. ⭐⭐⭐ A solid ball of radius $R$ has density $\rho_{\text{density}} = k\,\rho$ proportional to distance from the center. Find its total mass using spherical coordinates. (Compare to the hemisphere result in §32.13.)

34. ⭐⭐⭐ Find the moment of inertia $I_z = \iint_R (x^2 + y^2)\,\sigma\,dA$ of a uniform disk of radius $R$ and constant area-density $\sigma$ about its central axis. Express the answer in terms of the disk's total mass $M = \sigma \pi R^2$.

Part D — Synthesis ⭐⭐⭐⭐

These weave together several sections and at least two application fields.

35. ⭐⭐⭐⭐ (The Gaussian normalization — statistics.) Starting from $I = \int_{-\infty}^{\infty} e^{-x^2}\,dx$, reproduce the §32.5 derivation that $I = \sqrt{\pi}$ by squaring, converting to a double integral over $\mathbb{R}^2$, and using polar coordinates. Then use the substitution $x = \sqrt{2}\,t$ to show $\int_{-\infty}^{\infty} e^{-x^2/2}\,dx = \sqrt{2\pi}$, and conclude that $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ integrates to exactly $1$. State in one sentence why this number underlies every $z$-score in statistics.

36. ⭐⭐⭐⭐ (Center of mass of a machined part — engineering.) A flat bracket occupies the region between $y = x^2$ and $y = \sqrt{x}$ for $0 \le x \le 1$, with density increasing toward the right as $\rho(x,y) = 1 + x$ (denser material near the mounting bolt). (a) Find the mass $M$. (b) Find the moment $M_y = \iint x\,\rho\,dA$ about the $y$-axis. (c) Find $\bar x$. (d) Explain physically why $\bar x$ shifts to the right of the centroid of the same region under uniform density.

37. ⭐⭐⭐⭐ (Joint probability over a region — data science.) A pair $(X,Y)$ has joint density $p(x,y) = c\,xy$ on the unit square $[0,1]^2$ and $0$ elsewhere. (a) Find $c$ so that $\iint p\,dA = 1$. (b) Compute $P(X + Y \le 1)$ by integrating over the triangular sub-region. (c) Find the marginal density $p_X(x) = \int_0^1 p(x,y)\,dy$ and verify it integrates to $1$.

38. ⭐⭐⭐⭐ (Coordinate choice as strategy — physics.) A solid is bounded below by the cone $z = \sqrt{x^2 + y^2}$ and above by the sphere $x^2 + y^2 + z^2 = 8$. (a) Find its volume using spherical coordinates. (b) Set up — but do not evaluate — the same volume as a cylindrical integral, and write two sentences comparing the difficulty of the two setups. (c) Find the $z$-coordinate of the centroid, $\bar z = \frac{1}{V}\iiint z\,dV$, exploiting the symmetry to skip $\bar x$ and $\bar y$.

Hints

• 15, 16, 23: the inner antiderivative does not exist in elementary form in the given order — that is your signal to reverse (§32.3, Example 3).
• 24–28: whenever the region's boundary involves $x^2 + y^2$ or a disk, polar coordinates almost always win (§32.5).
• 32, 38: "above the cone, inside the sphere" gives constant $\rho$- and $\phi$-limits in spherical coordinates — the dream scenario of Example 8.
• 34: after switching to polar, the integrand becomes $r^2 \cdot r = r^3$; integrate $\int_0^R r^3\,dr = R^4/4$.
• 36, 37: mass/moment problems and probability problems are the same integral with different names — $\iint(\text{weight})\,dA$ — so the mechanics transfer directly (§32.10–§32.11).