Area and Work How does the area of the ellipse
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Chapter 15 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- A 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_forwardApplications of Integration: Volumes of Solids of Revolutionarrow_forwardCirculation on a half-annulus Consider the vector field F = ⟨y2, x2⟩on the half-annulus R = {(x, y): 1 ≤ x2 + y2 ≤ 9, y ≥ 0}, whose boundary is C. Find the circulation on C, assuming it has the orientation shown.arrow_forward
- Finding a center of mass Find the center of mass of a thin plateof density d = 3 bounded by the lines x = 0, y = x, and the parabolay = 2 - x2 in the first quadrant.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_forwardWork in a hyperbolic field Consider the hyperbolic force field F = ⟨y, x⟩ (the streamlines are hyperbolas) and the three paths shown in the figure. Compute the work done in the presence of F on each of the three paths. Does it appear that the line integral ∫C F ⋅ T ds is independent of the path, where C is any path from (1, 0) to (0, 1)?arrow_forward
- Finding Iz in spherical coordinates Find the moment of inertiaabout the z-axis of a solid of constant density d = 1 that is boundedabove by the sphere r = 2 and below by the cone f = π/3(spherical coordinates).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_forwardSurface 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_forward
- Engineering Mechanics - Centroids Using Centroid by Integration, determine the x- and y-coordinates of the centroid of the shaded area.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_forwardMultivariable calculus. Let F vector = <x,y,z> and use the Divergence Theorem to calculate the (nonzero) volume of some solid in IR3 by calculating a surface integral. (You can pick the solid).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