Double and Triple Integrals - Question Bank

1. 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.
A) 0
B) 15π/4
C) 15π/2
D) 15π
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.
A) (1/2)MR^2
B) MR^2
C) (1/4)MR^2
D) (2/3)MR^2
3. Find the volume of the solid bounded by the paraboloid z = 1 - x^2 - y^2 and the xy-plane.
A) π/2
B) π
C) 2π
D) 4π
4. 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.
A) 1/3
B) 5/12
C) 7/15
D) 3/4
5. 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.
A) 0
B) π/2
C) π
D) 2π
6. 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)?
A) 1/6
B) 1/3
C) 1/2
D) 1
7. Evaluate the double integral ∫∫_R e^(x+y) dA over the rectangle R = [0, 1] x [0, 1].
A) e^2 - 2e + 1
B) e^2 - 2e
C) e^2 - 1
D) e - 1
8. Consider the region R in the first quadrant bounded by x=0, y=0, and x+y=1. Evaluate ∫∫_R x dA.
A) 1/24
B) 1/12
C) 1/8
D) 1/6
9. 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?
A) 3/2
B) 2
C) 5/2
D) 3
10. Evaluate ∫∫∫_E dV where E is the unit sphere x^2 + y^2 + z^2 ≤ 1.
A) 4π/3
B) π
C) 4π
D) 2π
11. What is the volume of the region E bounded by z = 0 and z = 4 - x^2 - y^2?
A) 8π
B) 4π
C) 16π
D) 12π
12. 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.
A) 1/2
B) 1
C) 3/2
D) 2
13. Evaluate the triple integral ∫_0^1 ∫_0^1 ∫_0^1 xyz dV.
A) 1/16
B) 1/8
C) 1/4
D) 1/2
14. Evaluate the double integral ∫∫_D x dA where D is the disk x^2 + y^2 ≤ 4.
A) 0
B) π
C) 4π
D) 8π
15. Evaluate the double integral ∫_0^1 ∫_0^x y dy dx.
A) 1/12
B) 1/6
C) 1/4
D) 1/3
16. Consider the region R bounded by x=0, x=2, y=0, y=3. Evaluate ∫∫_R (x+y) dA.
A) 18
B) 12
C) 24
D) 36
17. What does the double integral ∫∫_R dA represent?
A) The area of the region R.
B) The volume of the region R.
C) The perimeter of the region R.
D) The centroid of the region R.
18. 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?
A) ρ^2 sin(φ)
B) ρ sin(φ)
C) ρ^2
D) sin(φ)
19. 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?
A) Spherical coordinates
B) Cylindrical coordinates
C) Cartesian coordinates
D) Parabolic coordinates
20. What is the region of integration E in spherical coordinates if 0 ≤ ρ ≤ 2, 0 ≤ φ ≤ π/2, and 0 ≤ θ ≤ π?
A) The upper-right octant of a sphere of radius 2.
B) The upper hemisphere of a sphere of radius 2.
C) The first octant of a sphere of radius 2.
D) A quarter sphere of radius 2.
21. 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?
A) ∫_0^1 ∫_0^(2π) ∫_0^1 (r^2) r dr dθ dz
B) ∫_0^(2π) ∫_0^1 ∫_0^1 r^2 r dr dθ dz
C) ∫_0^1 ∫_0^(2π) ∫_0^1 r^2 r dr dθ dz
D) ∫_0^(2π) ∫_0^1 ∫_0^1 r^2 dr dθ dz
22. 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?
A) Cylindrical coordinates
B) Spherical coordinates
C) Cartesian coordinates
D) Polar coordinates
23. 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:
A) Constants
B) Functions of the inner variables
C) Variables
D) Zero
24. Which theorem is analogous to Fubini's Theorem for triple integrals, allowing the computation as iterated integrals?
A) Fubini's Theorem for triple integrals
B) Green's Theorem
C) Stokes' Theorem
D) Divergence Theorem
25. The average value of a function f over a region E is given by:
A) (1 / Volume(E)) * ∫∫∫_E f dV
B) Volume(E) * ∫∫∫_E f dV
C) ∫∫∫_E f dV
D) (1 / SurfaceArea(E)) * ∫∫∫_E f dV
26. 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?
A) The total mass of the object.
B) The volume of the object.
C) The average density of the object.
D) The surface area of the object.
27. The triple integral ∫∫∫_E dV calculates:
A) The volume of the region E.
B) The mass of the region E.
C) The average density of E.
D) The surface area of E.
28. 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?
A) ∫_0^(2π) ∫_0^π ∫_0^R f(ρ, φ, θ) ρ^2 sin(φ) dρ dφ dθ
B) ∫_0^R ∫_0^π ∫_0^(2π) f(ρ, φ, θ) ρ^2 sin(φ) dρ dφ dθ
C) ∫_0^π ∫_0^(2π) ∫_0^R f(ρ, φ, θ) dρ dφ dθ
D) ∫_0^(2π) ∫_0^R ∫_0^π f(ρ, φ, θ) ρ^2 dρ dφ dθ
29. In spherical coordinates (ρ, φ, θ), the differential volume element dV is given by:
A) ρ^2 sin(φ) dρ dφ dθ
B) ρ dρ dφ dθ
C) r dr dθ dz
D) dx dy dz
30. What transformation is commonly used to simplify triple integrals over regions with spherical symmetry?
A) Spherical coordinates
B) Cylindrical coordinates
C) Polar coordinates
D) Cartesian coordinates
31. 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?
A) ∫_0^H ∫_0^(2π) ∫_0^R f(r, θ, z) r dr dθ dz
B) ∫_0^R ∫_0^(2π) ∫_0^H f(r, θ, z) r dr dθ dz
C) ∫_0^(2π) ∫_0^R ∫_0^H f(r, θ, z) dr dθ dz
D) ∫_0^H ∫_0^R ∫_0^(2π) f(r, θ, z) r dθ dr dz
32. In cylindrical coordinates (r, θ, z), the differential volume element dV is given by:
A) r dr dθ dz
B) dr dθ dz
C) r^2 sin(φ) dr dφ dθ
D) dx dy dz
33. What transformation is commonly used to simplify triple integrals over regions with cylindrical symmetry?
A) Cylindrical coordinates
B) Spherical coordinates
C) Polar coordinates
D) Cartesian coordinates
34. What is the Jacobian of the transformation from Cartesian (x, y) to polar (r, θ) coordinates?
A) r
B) 1/r
C) 1
D) sin(θ)
35. 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?
A) ∫_0^(2π) ∫_0^R f(r, θ) r dr dθ
B) ∫_0^R ∫_0^(2π) f(r, θ) dr dθ
C) ∫_0^(2π) ∫_0^R f(r, θ) dr dθ
D) ∫_0^R ∫_0^(2π) f(r, θ) r dr dθ
36. In polar coordinates, the differential area element dA is given by:
A) r dr dθ
B) dr dθ
C) r^2 sin(φ) dr dφ dθ
D) dx dy
37. What transformation is commonly used to simplify double integrals over non-rectangular regions, especially those involving circles or sectors?
A) Polar coordinates
B) Spherical coordinates
C) Cylindrical coordinates
D) Cartesian coordinates
38. 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?
A) ∫_0^1 ∫_0^x ∫_0^(x+y) x dz dy dx
B) ∫_0^1 ∫_0^y ∫_0^(x+y) x dz dx dy
C) ∫_0^x ∫_0^1 ∫_0^(x+y) x dz dy dx
D) ∫_0^1 ∫_0^1 ∫_0^1 x dz dy dx
39. When evaluating a triple integral ∫∫∫_E f(x, y, z) dV using iterated integrals, which order of integration is generally NOT possible?
A) dz dy dx
B) dx dz dy
C) dy dx dz
D) dxdydz
40. For a function f(x, y, z) = 1 over a region E, what does the triple integral ∫∫∫_E 1 dV represent?
A) The volume of the region E.
B) The surface area of E.
C) The average value of f over E.
D) The density of E.
41. If a triple integral is set up as ∫∫∫_E f(x, y, z) dV, what does 'dV' typically represent in Cartesian coordinates?
A) dx dy dz (in any order)
B) dx dy
C) dA
D) ds
42. What is the primary geometric interpretation of a triple integral over a region E in 3D space?
A) The volume of the region E.
B) The mass of the region E if the integrand is density.
C) The flux through a surface bounding E.
D) The work done by a force field over a path within E.
43. 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) ∫_a^b ∫_f1(x)^f2(x) g(x, y) dy dx
B) ∫_a^b ∫_f2(x)^f1(x) g(x, y) dy dx
C) ∫_f1(x)^f2(x) ∫_a^b g(x, y) dx dy
D) ∫_a^b ∫_a^b g(x, y) dy dx
44. 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?
A) ∫_0^1 ∫_0^x x y dy dx
B) ∫_0^x ∫_0^1 x y dx dy
C) ∫_0^1 ∫_x^1 x y dy dx
D) ∫_0^1 ∫_0^1 x y dy dx
45. What is the key condition for Fubini's Theorem to apply for evaluating double integrals over a rectangle?
A) The function f(x, y) must be continuous over the rectangle.
B) The function f(x, y) must be differentiable.
C) The function f(x, y) must be monotonic.
D) The function f(x, y) must be periodic.
46. 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:
A) ∫_c^d (∫_a^b f(x, y) dx) dy
B) ∫_a^b (∫_c^d f(x, y) dy) dx
C) Both of the above
D) Neither of the above
47. For a function f(x, y) = c (a constant) over a region R, what is the value of the double integral ∫∫_R c dA?
A) c times the area of R.
B) c times the perimeter of R.
C) c.
D) 0.
48. If a double integral is set up as ∫∫_R f(x, y) dA, what does 'dA' typically represent in Cartesian coordinates?
A) dx dy or dy dx
B) dr dθ
C) dz
D) dt
49. What is the primary geometric interpretation of a double integral over a region R in the xy-plane?
A) The volume under the surface z = f(x, y) and above the region R.
B) The area of the region R.
C) The arc length of a curve within the region R.
D) The surface area of the region R.
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