Evaluate the surface integral.     Sz2 dS, S is the part of the paraboloid  x = y2 + z2  given by  0 ≤ x ≤ 4

Q: 5. Consider the surface: x(u, v) y(u, v) z(u, v) where u € [0, 1] and v € [0,2]. Calculate the area…

A: Introduction: The area is the area occupied by a two-dimensional flat surface. It has a square unit…

Q: Calculate F· dS where F = (3xy', ze", 2°) and S is the surface of the solid bounded by the cylinder…

A: We need to calculate ∬SF→.dS which is the flux. Now, we can use the divergence theorem to find…

Q: Apply Green's Theorem to evaluate the integral. $(4y + x) +x)dx + (y + x)dy C: The circle (x-3)² +…

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Q: 1. Use Green's Theorem to evaluate the line integral (y + lnx) dx - x²dy over the rectangle in the…

A: (.)  Given line integral is ,             ∮c(y+lnx)dx-x2dy where  c is the rectangle in xy plane…

Q: Determine the surface area of the plane 2x+y + 2z = 16 cut off by x = 0, y = 0 and x² + y² = 64.

A: Area of surface given by z = f(x,y) A=∫∫Dfx2+fy2+1dA  where (x,y) is a point from the region D in…

Q: Find the surface area of that portion of the surface z = V4 -x that lies above the rectangle R in…

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Q: Find the equation that solves the area of the surface obtained by rotating the curve x = (y² + 2)% ,…

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Q: Let C be the positively oriented square with vertices (0,0), (1, 0), (1, 1), (0, 1). Use Green's…

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Q: Find the area of the surface. The part of the hyperbolic paraboloid z = y² - x² that lies between…

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Q: Consider the curve y = V 2x – x² where 0 < x < 2. -

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Q: Use Green's Theorem to evaluate the following integral Let² dx + (5x + 9) dy Where C is the triangle…

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Q: Evaluate the surface integral. x ds, S is the triangular region with vertices (1, 0, 0), (0, –2, 0),…

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Q: 2. Let D be the parallelogram bounded by the lines x+3y = 5, –x + y = 3, –x + y = 2, and x + 3y = 0.…

A: Let us consider the integral I=∫∫Df(x,y) dydx where D is the region in xy-coordinate  To make the…

Q: S is the part of the surface z = x2 + y2 + 4 that lies within the cylinder x2 + y2 = 9. Find the…

A: Given The surface z=x2+y2+4that lies within the cylinder x2+y2 = 9.

Q: 3. Evaluate the integral using polar coordinates. So Sy=z=1/² √² + y² dydz √1-(2-1)²

A: To find : ∫02∫1-x-124-x2 xx2+y2dy dx

Q: Uv Use the change of variable x = and y in order to v+ 4 v + 4 compute the integral 4.x + y dA. D is…

A: Here we solve the integral problem.

Q: What is the equation of the surface revolved about the x-axis for z² - xz = ln x + 3x? A. y² + z² =…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly.…

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Q: Evaluate the surface integral

A: Given,                ∫∫S8z2dS Where S is the boundary of the cube C=-1, 1×-1, 1×-1, 1.

Q: 5. Find the surface area of the part of the paraboloid y = 5– x² – 2² that lies inside the cylinder…

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Q: 1. Find the smallest distance from the point (0, c, 0) to the surface y = x² + z².

A: We have to use Langrages Multipliers Method to find the minimum distance from (0,c,0) to a point on…

Q: Exer.) Express and evaluate the integral (x+y) dv E as an iterated integral for the given solid…

A: Given:

Q: Q.2: Let C be the line joining A(-1,0), B(0,1), D(1,1), and E(2,0). Evaluate: Scy dz tor and radius…

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Q: Find the point on the surface r – 2xy + 2y² – x+ y = 0 closest to the point (1, 2, –3).

A: Consider the provided question, We have to find the point on the given surface closest to the point…

Q: EX. Find the volune of the solid region lymg mside both the sphere Xtuf+2?= 402 and the cylmeder…

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Q: 2. Find the exact area of the surface. a. z = 1+ 2x + 3y + 4y², 1 < x < 4, 0 < y < 1. b. z = 1+x +…

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Q: Find div F and curl F if F(x, y, z) = x*i – 11j + yzk. div F= curl F=

A: Vector