Moments of Inertia Find the moments of inertia for a wire
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Chapter 15 Solutions
Calculus: Early Transcendental Functions
- Surface Integral over a Graph Calculate the flux of F = x2j through the surface S defined by y = l + x 2 + z2 for 1 < y < 5. Orient S with normal pointing in the negative y-direction.arrow_forwardMass from density data The following table gives the density (in units of g/cm2) at selected points (in polar coordinates) of a thin semicircular plate of radius 3. Estimate the mass of the plate and explain your method.arrow_forwardA thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2x in the first quadrant. The plate’s density at the point (x, y) is δ(x, y) = 6x + 6y + 6. Find the plate’s moments of inertia about the coordinate axes and the origin.arrow_forward
- Work 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_forwardA thin plate occupies region D of the plane bounded by the curves y = 1 - x^2 and y = 4 - 4x^2. The density of the plate is proportional to the square of the distance to the origin. Determine the center of mass of the platearrow_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
- *INTEGRAL CALCULUS Show complete solution (with graph). 5. Determine the centroid of the solid generated by revolving the area bounded by the curve y = x^2, y = 9, and x = 0, about the y − axis.arrow_forwardA lamina occupies the part of the disk x2 + y2 ≤ 16 in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.arrow_forwardA thin plate of constant density δ = 1occupies the region enclosed by the curve y = 36/(2x + 3) andthe line x = 3 in the first quadrant. Find the moment of the plateabout the y-axis.arrow_forward
- Kinetic energy of a fluid flow can be computed by ∭V12ρv⋅vdV∭V12ρv⋅vdV, where ρ(x,y,z)ρ(x,y,z) and v(x,y,z)v(x,y,z) are the pointwise fluid density and velocity, respectively. Fluid with uniform density 7π7π flows in the domain bounded by x2+z2=7x2+z2=7 and 0≤y≤670≤y≤67. The velocity of parabolic flow in the given domain is v(x,y,z)=(7−x2−z2)j⃗ v(x,y,z)=(7−x2−z2)j→. Find the kinetic energy of the fluid flow.arrow_forwardA trough is full of liquid of weight density 9,000 N/m3. The ends of the trough are equilateral triangles with sides 2 m long and with vertex at the bottom. Setup the integral for the force on one end of the trough. (Hint: Put the bottom of the trough along the x-axis.)arrow_forwardAverage circulation Let S be a small circular disk of radius Rcentered at the point P with a unit normal vector n. Let C be theboundary of S.a. Express the average circulation of the vector field F on S as asurface integral of ∇ x F.b. Argue that for small R, the average circulation approaches(∇ x F) | P ⋅ n (the component of ∇ x F in the direction of nevaluated at P) with the approximation improving as R → 0.arrow_forward
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- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning