(1) Find the area of the curved surface of the upper half of the sphere x2 + y² + z² = a²(a > 0) on a loop of the rose line r = a cos 20 in the xy plane.

Q: Show that the curve r= Vti + Vtj + (2t – 1)k is tangent to the surface x +y² – z = 1 when t=1

A: Given that r=ti+tj+2t-1k and f=x2+y2-z-1 Take the derivative for f, ∇f=ddx,ddy,ddzf=2x,2y,-1…

Q: 24. Evaluate | x/x² + y² dA, where D is the shaded region enclosed by the lemniscate curve r² = sin…

A: Consider the given:

Q: Find the arclength of the graph of the fun ction over the indicated inter val. (a) y = 2x3/2 + 3, 0…

A: Given : Functions and their interval (a) y = 2x3/2 + 3 , 0≤x≤9 (b) y = ln(cosx), [0, π3] (c) x =…

Q: 18. Show that the centroid of the sector in Figure 13 has y-coordinate 2R sin o (0, y) R FIGURE 13

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Q: Find the area of the part of the eircle r-4 sin that lies within the sector 0 50s 87 -2/3 3 + V3 3.…

A: It is given that,r=4sinθ, that lies within the sector 0≤θ≤π3we have to find the area of the part of…

Q: 6. Evaluate the line integral g. x²y² dx + xy dy where C consists of the arc of the parabola y = x²…

A: (a) Direct method The given line integral is ∮C x2 y2 dx + xy dy. The curve C consists of    C1 :…

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

A:

Q: Find the area of the surface generated by eY + e -y revolving the curve x = in the interval -In 2…

A:

Q: Use Green's Theorem to evaluate the integral (y - x)dx + (2x - y)dy for the given path. C: x = 2…

A: Definition used -                                According to the green theorem -…

Q: Use Green's Theorem to evaluate the line integral |(y - x)dx + (2x – y)dy for the given path. C: x =…

A: Let's find.

Q: 1. Find by integration the area of the surface of revolution generated by revolving the loop about…

A:

Q: 4. Graph the surface 1 = y² + z² – by first sketching at least three cross sections in each of the…

A: Graph the surface by using a 3D graphing calculator like desmos. Three cross sections in xy plane…

Q: (7.5.0)- Find an equation of the tangent plane to the surface of z = sin(x – 2y) at the point %3D

A: Given F(x, y, z)=z-sin(x-2y) ∂∂xF(x, y, z)=∂∂x(z-sin(x-2y))=-cos(x-2y)∂∂yF(x, y,…

Q: Jse Green's theorem to evaluate the ne integral (2tan x _cos 2y³) dx + (7⁰s y + 2x (πº where is the…

A:

Q: Find the area of the surface formed by revolving the graph of f(x) = x3 on the interval [0, 1] about…

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Q: The area of the surface obtained by rotating the curve y = 1<xse about the line x = 32 is ** 12 (32…

A:

Q: The area of the surface obtained by rotating the curve y = 1 I+ X, 1 <xse about the line x = 28 is…

A:

Q: 4. Graph the surface 1 = y² + z² by first sketching at least three cross sections in each of the 4…

A: We have to graph the surface 1=y2+z2-x24 by sketching at least three cross-sections in each of the…

Q: The surface area of the surface generated by revolving y=1-x² for 0<x<1 about the x- axis is s=…

A: If you like the solution then please give it a thumbs up.... The Answer is:              S =…

Q: Use Green's Theorem to evaluate the line integral Jo (y - x)dx + (2x - y)dy for the given path. C: x…

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

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Q: 12. Find the area of the surface generated when the curve y = ,on [1,2] is rotated about the x-axis.

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Q: Use Green's Theorem to evaluate the line integral √(x - (y - x)dx + (2x - y)dy for the given path.…

A: Let's find.

Q: Use Green's Theorem to evaluate the integral. Assume that the curve C'is oriented counterclockwise.…

A: This is a problem related to vector calculus. Based on the general formula of green's theorem we…

Q: FD is the region in the first octant between the spheres x2+y2+z²=1 and x2+y2+z²=16, then dV =

A: Converting into spherical coordinates

Q: Use Green's Theorem to evaluate the line integral. (e-x²12-v) dx + (0-1²/2 + x) ay dy C: boundary of…

A:

Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

A: Given integral is ∫Cy−xdx+2x−ydy where C:x=2cosθ,y=sinθ,0≤θ≤2π. Green theorem is given by…

Q: Compute the integral „2020 (5y + x*0 ") dx + (19x – e -e-4yCos(y) sin y) dy over the portion of the…

A:

Q: 1. Determine the centroid of the first full enclosed region in the first quadrant between sin r and…

A: given curves are                                              f(x)=sinxg(x)=cosx graph of the…

Q: Find the area of the surface generated by e +e -y eY + e -y in the X= revolving the curve x= -In 3…

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Q: 3. Find the exact area of the surface resulting from rotating y = v1 + e* from x = 0 to x = 1 about…

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Q: Integrate ƒ(x, y, z) =-2x2 + z2 over the circle r(t) = (a cos t)j + (a sin t)k, )  0&lt;=…

A:

Q: Use Green's Theorem to calculate 4. [(x² – xy) dx + (y² - 2xy) dy] where C is a square with vertices…

A:

Q: The surface area of the surface generated by revolving y= 1-x2 for 0<x<1 about the x - axis is S = […

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Q: 1. Find the area of the surface generated by revolving the given curve about the x-a (a) y = 7x,…

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Q: The area of the surface obtained by rotating the curve y = -+ X°, 1sx< e about the line x = 24 is x3…

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Q: Evaluate |(2y+cos x)dx + (5x+ sin y)dy along the positively oriented curve C. C is the boundary of…

A: We have to evaluate the integral ∫2y+cosxdx+5x+sinydy along the positively oriented curve C. Where C…

Q: y = cos(x), 0 <x<

A: Consider the given surface whose area is required to calculate.

Q: The area of the surface obtained by rotating the curve y = 1 < x s e about the line x = 24 is O 2n…

A:

Q: Find the area of the surface generated by eY + e -Y revolving the curve x = in the interval 2 In 5…

A: solution:-given curve x=ey+e-y2 in the interval 0≤y≤ln5revolving about the y-axisarea of a surface…

Q: (a) Calculate the centroid (7, y) of the region under the graph y = cos x for 0 < x <T/2.

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

Q: (A) Find (jx curlū), grad (yzj • curlū) at น 7 = ({√Cos(1³² - x)) + ( x + ² } / + 20² k ze² where

A: Given u=cosy2-xi+y+1zj+zexzk

Q: 13. Find the area of the surface generated when the curve = , on [2,4] is rotated about the y-axis.

A: The solution are next step is

Q: 7. Evaluate the integral I | sin(9.r² + 4y²)dA by hand, using an appropriate change of variables,…

A: The given question can be solved as shown in step2

Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A:

Q: 12. Find the area of the surface generated when the curve y =+,on [1,2] is rotated about the x-axis.

A: solution is given below...

Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A: First we have to find partial differentiation with respect to x , y and z of function f such that-

Q: 3.4.6 Evaluate the line integral [sin y dx + (x- cos y) dy] taken in the anticlockwise sense, where…

A: Green's Theorem States that  Let C be the positively oriented, smooth, and simple closed curve in a…

Q: 1. (a) Find the exact area of the surface obtained by rotating the curve y = e* about the x-axis…

A: