Consider a point (X, Y) E R² with joint density for some C≥ 0: f(x, y) = Ce (x, y) = R². x²+y (a) Find the constant C. (b) Let (R, O) be the polar coordinates of (X, Y). Find the density of R, O. Show that R, are independent. (c) What is the probability that (X, Y) lands on the 3rd quadrant? (d) Find the expected value and variance of R. (e) Let (X', Y') be another point on R2 with density f, which is independent of (X, Y). Find the probability that there is at least one of the point is landed in the ball B₁(0) of radius 1 centered at origin (0,0).

Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

A:

Q: Let F = (y², x, z²)and S be the part of the paraboloid z = x² + y² with z < 1 and outward facing…

A:

Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

A: The surface area of the helicoid can be computed using the surface integral formula:   ∫∫Rru×rvdA…

Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) (u cos v, u sin v,…

A:

Q: find laplace of f(x,y,z)=2x3y2z2

A:

Q: Let T be the triangular region region with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1) oriented…

A:

Q: F. Sketch the region onto which the sector r ≤ 1,0 ≤ 0 ≤/4 is mapped by the transfor- mation (a) w =…

A:

Q: Find the flux of F across S, 1.1. S F. N dS where N is the upward unit normal vector to S. F(x, y,…

A:

Q: 2. Calculate the Jacobian of the coordinate change (x, y, z) to cylindrical coordinates (r, 0, z).

A: To calculate the Jacobian of the coordinate change  to cylindrical coordinates .

Q: 2. Find the directional derivative of f(x,y, z) = x²y + yz² at P(1,–1,3) in the direction of vector…

A: