Let R be the region of the xy-plane which is in the first quadrant and inside the unit circle x² + y = 1 In other words, R = {(x, y) | x > 0, y 2 0 and x² + y° < Find the x-coordinate of the center of mass of R

Q: Let G = {(x, y): 1 ≤ x^2 + y^2 ≤ 16,0 ≤ y ≤ x). Sketch G, and find the coordinates of its centroid.

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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: The distance of a point P(x,y,z) from the xy plane is d = |x| how do we show the work to prove this?

A: To show the distance of a point P(x, y, z) from the xy-plane is d = |x|, it can use the distance…

Q: A conical domain with vertex (0, 0, b) and axis along the z-axis has as base a disk of radius a in…

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Q: Let S be the portion of the surface z=Vy²+1+e* that lies above the region R= {(x,y)ER?:x²+y?s1,y20}…

A: Given the surface is z=y2+1+ex,and R=x,y∈ℝ2|x2+y2≤1,y≥0. Find I=∬SF·ndσ, where F=ye-x,2ex-2z,y.

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

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Q: et Q be the part of the sphere of radius 3 centered at the origin which lies above z = 1, and let D…

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

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Q: Find the flux of G(x, y, z) = (xz)i+ (yz)j+(z²)k outward across the part of the sphere x² + y² +z² =…

A: Here, surface, S={(x,y,z): x2+y2+z2=9, and 1≤z≤2}. For z=1, we get x2+y2=8 and for z=2, we get…

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,…

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Q: Compute the flux of F = xi + yj + zk through just the curved surface of the cylinder x² + y² = 16…

A: We have to compute the flux of F=xi+yj+zk through the curved surface of the cylinder x2 + y2 = 16…