Explain how to compute the surface integral of a scalar-valued function ƒ over a sphere using a parametric description of the sphere.

Q: Does the Divergence Theorem apply to surfaces that are not closed?

A: By Stokes' theorem:

Q: State the theorem for Vector Surface Integral.

A: More precisely, the divergence theorem states that the surface integral of a vector field over a…

Q: 50.) Show how to use a line integral to find the area of a circular disk of radius r

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Q: FIND THE CENTROID OF THE AREA BOUNDED BY THE CURVE r = a (1 + sin 0).

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Q: Write The Green's Theorem in the plane relating double integrals and line integrals. (Please, don't…

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

Q: Find the area of the triangle with vertices (0,0), (2, 0) and (2,2) by computing a double integral…

A: As per the question we are given a (right-angled) triangle with vertices at (0,0), (2,0), (2,2) And…

Q: The Green's Theorem allows us to replace any line integral over simple, smooth, closed curve in…

A: Yes, Green' Theorem allows us to replace any line integral over simple smooth closed curve in 3D…

Q: Calculate the y-coordinate of the centroid of the shaded area.

A: Figure shows the cross section of the composite section: Divide the composite section into figure…

Q: please describe clearly, and type by computer  Describe how to set up a surface integral given that…

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Q: Use Stokes' Theorem to find the line integral of F = 3yi + 2zj + 5xk around the triangle obtained by…

A: Given function is F→=3yi^+2zj^+5xk^. We have to evaluate the line integral of F→ around the triangle…

Q: Find the centroid of the region that is enclosed between: y = x and y = 2 – 2?.

A: Let fx=2-x2, gx=x. Find the intersections. fx=gx2-x2 =x2-x2…

Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices…

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Q: Find the area and centroid bounded by y= 2(1-x^2), y=x^2-1. Graph.

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Q: Calculate the scalar line Integral where the function is f(x,y)= 1 / x²+y²+1 and the curve is the…

A: We have given function fx,y=1x2+y2+1 and the curve joining the point 0,0 1,0 1,1 0,1. Let A0,0,…

Q: Find the average distance of a point in a disk of radius a to a fixed point on the boundary of the…

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Q: Determine the centroid of the lamina bounded by y=sinx, y=cosx and x=0 in the interval 0,

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Q: Express the operators ( gradient , divergence curl and laplacian ) in spherical coordinates.

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Q: 5 Compute the Surface area f obtained by revoluing the line y=x around the y-aris. a cone whichü

A: To find the surface area of cone which is obtained by revolving y=x revolved around y-axis.

Q: Explain the meaning of the surface integral in the Divergence Theorem.

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Q: Find the centroid of the homogeneous lamina bounded by the graph whose borders are f(x) = 16 − x2…

A: Let k be the density of the lamina body which is bounded by f(x) and (x-axis)

Q: Example Find the exact area of the surface obtained by rotating the curve y = v1+ ex, 0<x< 1, about…

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Q: Find the centroid of a thin, flat plate covering the region enclosed by the parabolas y = 2x2 and y…

A: Given,

Q: Find the Integral Centroid of Plane Area Under a Curve

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Q: Evaluate the spherical coordinate integral.

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Q: Determine the area of the surface of revolution defines by the function f(x) = Vx,1< x< 4 %3D

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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: Find a parametrization of the curve. The intersection of the surfaces z = x^2 − y^2 and z = x^2 + xy…

A: Given : The surfaces z = x2 - y2 and z = x2 + xy - 1.   To find the parameterization of the curve.

Q: Find the centroid of a lamina enclosed between the circles of radius 1 and 2 centred at the origin,…

A: Consider the diagram of the lamina as per given information, The centriod of the lamina is defined…

Q: Sketch the region bounded by the given curves and find its centroid

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Q: Let f:[a,b] is Riemann integr only if it is Riemann integr

A: We have to solve given problem:

Q: 2.Proof that any tangent plane for the surface F(- point = 0 passses through a fixed

A: Given below clear explanation

Q: Explain how to set up a surface integral given that you will project the surface onto the xz-plane.

A: Let the provided surface is S and the function is f whose integral over surface S is to be find.…

Q: 2. Use a parametrization to express the area of the given surface as a double integral and then…

A: Given ,x2+y2+z2 = 22       _________(I)and z = x2+y2     __________(II)Putting the value of z from…

Q: 4+ g(x) = x + 2 1 2 3.

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Q: Set up and evaluate the definite integral for the area of surface generated by revolving the curve…

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Q: Let R be the disc of radius 1 centered at the origin, and let f(x, y) be a function. Express SSR…

A: Explanation of the answer is as follows

Q: Use Cauchy-Riemann equations to find all points z such that fis differentiable: (a) f(z)= (b)…

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Q: Find the centroid of the plane area bounded by the parabola y = 6x - x² and the line y = x

A: To Find: The centroid of the plane area bounded by the parabola y=6x-x2 and the line y=x

Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices 0,…

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Q: Use Pappus’s Theorem for surface area and the fact that the sur-face area of a sphere of radius a is…

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Q: Find the centroid of the semicircular area inside x2+ y2 = a2 with y ≥ 0. Show the graph for it.

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Q: 12. Informally explain how a surface area generated by a curve of rotation can be computed with an…

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Q: Find the centroid of a thin flat plate covering the region enclosed by the parabolas y = 2x and y =…

A: Solution of question is written in step2 and step3.

Q: Evaluate the surface integral.

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Q: How to calculate the pseudo-surface integral from a line? Please explain it in your multivariable…

A: How to calculate surface integral from line?

Q: What is a parametrization of a curve in the xy-plane? Does a function y = ƒ(x) always have a…

A: Parametrization of a curve in the xy-plane:Parametrization of a curve consists of both equation and…

Q: タ=kr

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Q: Determine the area and centroid bounded by the curves: y = 6x – x² and y = x? – 2x.

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Q: Find the coordinates of the centroid of the region bounded by the straight line y = (2/π)x and the…

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Q: Write a double integral that represents the surface area of the portion of the plane z = 3 that lies…

A: Given: A double integral that represents the surface area of the portion of the plane z = 3 that…