Sketch the region bounded by the curves x=y34,x=2, and y=0 then find the volume of the solid generated by revolving this region about the y-axis.
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- A particle moves along line segments from the origin to the points (2, 0, 0), (2, 5, 1), (0, 5, 1), and back to the origin under the influence of the force field F(x, y, z) = z2i+ 3xyj + 3y2k. Use Stokes' Theorem to find the work done.The linear kos function is written by the equation Y = a + bX. of the equation, X represents.....What is the Riemannian metric? A) A function that defines the inner product of tangent vectors at each point on a manifold B) A function that measures the curvature of a surface C) A function that defines the volume of a manifold D) A function that determines the Euler characteristic of a manifold
- Please show all work ง 5. Evaluate the vector surface integral JJ (+2², x + 2y + 32², 5x − 2y + 2³) . Ndo. S Here, S is the sphere Sy²+2 1. = Since S is a closed surface, the outside of S is considered as the positive side on S. Hint: use the divergence theorem.Draw a graph of return on y-axis and standard devistion on x-axisA system is described by the action S[r, r'] = [ dt (x² + 4rª + mr³4), r(t) where m is a constant. Find the Euler-Lagrange equation for r = which extremises S. Interpret each of the terms that appear in the Euler- Lagrange equations in terms of Newton's second law. Find a conserved quantity for this system.
- Draw the graph of return on the y-axis and standadr deviation on the x-axisUse the given conditions to write an equation for the line in point-slope form and in slope-intercept form.Find at least the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation. y'' +(x+6)y'+y=0 y(x) = ☐ + ... (Type an expression in terms of a and a₁ that includes all terms up to order 3.)
- Let f be the function defined as, Ր f(x) = [* sin (t³) dt. Determine an expression for f(x) as a power series.Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. (Use S1, S2, and S3, respectively, for the vectors in the set.) (0, 0) = S={(5, 2), (-1, 1), (4, 0)) Express the vector s₁ in the set as a linear combination of the vectors $2 and $3. 51= Need Help? Read It Watch It Submit AnswerFind the derivative of the function f by using the rules of differentiation. 7 f(x) = -X417 4 f'(x) =

