Double and Triple Integrals - One Line Questions

1. The average value of a function f over a region E is given by: — (1 / Volume(E)) * ∫∫∫_E f dV
2. What is the moment of inertia of a solid disk of radius R and mass M about its central axis (perpendicular to the disk)? Assume uniform density. — (1/2)MR^2
3. If a region R is described in polar coordinates by 0 ≤ r ≤ R and 0 ≤ θ ≤ 2π (a disk of radius R), what is the double integral of f(r, θ) over R? — ∫_0^(2π) ∫_0^R f(r, θ) r dr dθ
4. If a region E is described in spherical coordinates by 0 ≤ ρ ≤ R, 0 ≤ φ ≤ π, and 0 ≤ θ ≤ 2π (a sphere of radius R), what is the triple integral of f(ρ, φ, θ) over E? — ∫_0^(2π) ∫_0^π ∫_0^R f(ρ, φ, θ) ρ^2 sin(φ) dρ dφ dθ
5. If the region of integration E is defined by x^2 + y^2 ≤ 1 and 0 ≤ z ≤ 1, how would you set up ∫∫∫_E (x^2 + y^2) dV in cylindrical coordinates? — ∫_0^1 ∫_0^(2π) ∫_0^1 r^2 r dr dθ dz
6. Consider the region E defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ x, and 0 ≤ z ≤ x+y. How would you set up the triple integral for ∫∫∫_E x dV? — ∫_0^1 ∫_0^x ∫_0^(x+y) x dz dy dx
7. Consider the region R defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ x. How would you set up the iterated integral for ∫∫_R x y dA? — ∫_0^1 ∫_0^x x y dy dx
8. If a region E is described in cylindrical coordinates by 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, and 0 ≤ z ≤ H (a cylinder), what is the triple integral of f(r, θ, z) over E? — ∫_0^H ∫_0^(2π) ∫_0^R f(r, θ, z) r dr dθ dz
9. If a region R is defined by y = f1(x) and y = f2(x) for a ≤ x ≤ b, with f1(x) ≤ f2(x), how is the double integral ∫∫_R g(x, y) dA typically set up? — ∫_a^b ∫_f1(x)^f2(x) g(x, y) dy dx
10. When evaluating a double integral ∫∫_R f(x, y) dA where R is a rectangular region [a, b] x [c, d], Fubini's Theorem allows us to compute it as: — Both of the above
11. Evaluate the double integral ∫∫_D x dA where D is the disk x^2 + y^2 ≤ 4. — 0
12. Evaluate the triple integral ∫∫∫_E y dV where E is the region bounded by the cylinder x^2 + y^2 = 1 and the planes z = 0 and z = 1. — 0
13. Evaluate ∫∫∫_E x dV where E is the region bounded by the spheres x^2+y^2+z^2=1 and x^2+y^2+z^2=4. — 0
14. Evaluate the double integral ∫_0^1 ∫_0^x y dy dx. — 1/12
15. Evaluate the triple integral ∫_0^1 ∫_0^1 ∫_0^1 xyz dV. — 1/16
16. Evaluate the triple integral ∫∫∫_E z dV where E is the region bounded by z=0, z=1, x=0, x=1, y=0, y=1. — 1/2
17. Consider the region R in the first quadrant bounded by x=0, y=0, and x+y=1. Evaluate ∫∫_R x dA. — 1/24
18. Evaluate the triple integral ∫∫∫_E dV where E is the region bounded by y=x^2, y=1, x=0, and z=0, z=x+y. — 5/12
19. What is the volume of the region under the plane z = x + y and above the triangle with vertices (0,0), (1,0), and (0,1)? — 1/6
20. Consider the region R bounded by x=0, x=2, y=0, y=3. Evaluate ∫∫_R (x+y) dA. — 18
21. What is the mass of a solid object occupying the region E = [0, 1] x [0, 1] x [0, 1] with density function ρ(x, y, z) = x + y + z? — 3/2
22. Evaluate ∫∫∫_E dV where E is the unit sphere x^2 + y^2 + z^2 ≤ 1. — 4π/3
23. What is the volume of the region E bounded by z = 0 and z = 4 - x^2 - y^2? — 8π
24. For a function f(x, y) = c (a constant) over a region R, what is the value of the double integral ∫∫_R c dA? — c times the area of R.
25. When setting up iterated triple integrals, the bounds for the inner integral can depend on the variables of the outer integrals, but the bounds for the outer integrals must be: — Constants
26. What transformation is commonly used to simplify triple integrals over regions with cylindrical symmetry? — Cylindrical coordinates
27. Consider a region E bounded by the surfaces z = 0, z = 4 - x^2 - y^2. To evaluate ∫∫∫_E z dV, which coordinate system would be most advantageous? — Cylindrical coordinates
28. If a triple integral is set up as ∫∫∫_E f(x, y, z) dV, what does 'dV' typically represent in Cartesian coordinates? — dx dy dz (in any order)
29. If a double integral is set up as ∫∫_R f(x, y) dA, what does 'dA' typically represent in Cartesian coordinates? — dx dy or dy dx
30. When evaluating a triple integral ∫∫∫_E f(x, y, z) dV using iterated integrals, which order of integration is generally NOT possible? — dxdydz
31. Evaluate the double integral ∫∫_R e^(x+y) dA over the rectangle R = [0, 1] x [0, 1]. — e^2 - 2e + 1
32. Which theorem is analogous to Fubini's Theorem for triple integrals, allowing the computation as iterated integrals? — Fubini's Theorem for triple integrals
33. What transformation is commonly used to simplify double integrals over non-rectangular regions, especially those involving circles or sectors? — Polar coordinates
34. What is the Jacobian of the transformation from Cartesian (x, y) to polar (r, θ) coordinates? — r
35. In polar coordinates, the differential area element dA is given by: — r dr dθ
36. In cylindrical coordinates (r, θ, z), the differential volume element dV is given by: — r dr dθ dz
37. What transformation is commonly used to simplify triple integrals over regions with spherical symmetry? — Spherical coordinates
38. To evaluate the integral ∫∫∫_E sqrt(x^2 + y^2 + z^2) dV where E is the unit ball (x^2 + y^2 + z^2 ≤ 1), which coordinate system is best? — Spherical coordinates
39. What does the double integral ∫∫_R dA represent? — The area of the region R.
40. What is the key condition for Fubini's Theorem to apply for evaluating double integrals over a rectangle? — The function f(x, y) must be continuous over the rectangle.
41. If f(x, y, z) represents the density of a solid object occupying region E, what does the triple integral ∫∫∫_E f(x, y, z) dV represent? — The total mass of the object.
42. What is the region of integration E in spherical coordinates if 0 ≤ ρ ≤ 2, 0 ≤ φ ≤ π/2, and 0 ≤ θ ≤ π? — The upper-right octant of a sphere of radius 2.
43. What is the primary geometric interpretation of a triple integral over a region E in 3D space? — The mass of the region E if the integrand is density.
44. For a function f(x, y, z) = 1 over a region E, what does the triple integral ∫∫∫_E 1 dV represent? — The volume of the region E.
45. The triple integral ∫∫∫_E dV calculates: — The volume of the region E.
46. What is the primary geometric interpretation of a double integral over a region R in the xy-plane? — The volume under the surface z = f(x, y) and above the region R.
47. Find the volume of the solid bounded by the paraboloid z = 1 - x^2 - y^2 and the xy-plane. — π/2
48. In spherical coordinates, the transformation equations are x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), and z = ρ cos(φ). What is the Jacobian of this transformation? — ρ^2 sin(φ)
49. In spherical coordinates (ρ, φ, θ), the differential volume element dV is given by: — ρ^2 sin(φ) dρ dφ dθ
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