Moments of Inertia In Exercises 59 and 60, set up a triple
Want to see the full answer?
Check out a sample textbook solutionChapter 14 Solutions
Multivariable Calculus
- Moment of inertia about y-axis of a square plate with surface density f(x, y)= k = constant and vertices with coordinates (0,0), (0.2), (2.0) and (2.2) points ? 14 k 3 A) 16 k 3 В) 13 -k 3 10 D) 3 11 E) 3arrow_forwardExercise 3 TRIPLE INTEGRAL IN CYLINDRICAL COORDINATES. Consider the region W that lies between the sphere x2 +y? + z? = 4, above the plane z = 0, and inside the cylinder a2 + y? = 1. (i) Sketch the region W. (ii) Use cylindrical coordinates to integrate f(r, y, z) = z over W.arrow_forwardUseDivergence theor em to find the ouward flux of F = 2xz i - 3xy j - zk across the boundary of theregion cut from the first octant by the planey +z = 4 and the elliptical cy linder 4x +y = 16. %3Darrow_forward
- 人工知能を使用せず、 すべてを段階的にデジタル形式で解決してください。 ありがとう SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE CHATGPT Find the moment of inertia about the x axis of a plate of constant density o = 1 limited by the circle x^2 + y^2 = 4. Use the result to find Ix and ly for the same plate. A sheet occupies part of the disk x^2 + y^2 ≤ 1 in the first quadrant. Find your center mass if the density at each point is proportional to its distance from the x-axis.arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction. F = ⟨y, z - x, -y⟩; S is the part of the paraboloidz = 2 - x2 - 2y2 that lies within the cylinder x2 + y2 = 1.arrow_forwardMaximum surface integral Let S be the paraboloidz = a(1 - x2 - y2), for z ≥ 0, where a > 0 is a real number.Let F = ⟨x - y, y + z, z - x⟩. For what value(s) of a (if any)does ∫∫S(∇ x F) ⋅ n dS have its maximum value?arrow_forward
- Using double integral in polar coordinates, find the area of the plane figure bounded by the curves x² – 2x + y2 = 0,x² – 4x + y² = 0,y = V3 ,y = V3 x.arrow_forwardWork by a constant force Evaluate a line integral to show thatthe work done in moving an object from point A to point B in thepresence of a constant force F = ⟨a, b, c⟩ is F ⋅ AB.arrow_forwardulus III |Uni Use Green's Theorem to evaluate the line integral cos (y) dx + x²sin (y) dy along CoS the positively oriented curve C, where C is the rectangle with vertices(0,0), (4, 0), (4, 2) and (0, 2).arrow_forward
- Use Stokes' Theorem to evaluate (integral) F (dot) dr. C is oriented counterclockwise as viewd from above. F (x,y,z) = < 2z + x , y - z , x + y > C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1).arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction.arrow_forwardKinetic energy of a fluid flow can be computed by ||| 5P pv · vdV, where p(x, y, z) and V 1 v(x, y, z) are the pointwise fluid density and velocity, respectively. Fluid with uniform density 6 flows in the domain bounded by x² + z² = 8 and 0 < Y< . The velocity of parabolic flow in 8 the given domain is v(x, y, z) = (8 – x² – z²)j. Find the kinetic energy of the fluid flow. Answer:arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning