d) R1 rotates around the y-axis e) Rp rotates around the y axis f) R3 rotates around the y axis y = Vx R, R. y = x/4

Q: 2. Use Green's Theorem (and only Green's Theorem) to evaluate |(2? + y)dx + (x² – y³)dy, where C is…

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A: Given, ∫∫∫Ey2z2dV   ....(i)Let x=ρsinϕcosθ y=ρsinϕsinθ and z=ρcosϕAnd we know…

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A:

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A: Here we have to find the Parameterization of path C, Counterclockwise around the circle x2+y2=1 from…

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A: To find the equation for the largest ball centered at (0, 0, 1) that can fit in the below equation…

Q: If KN=2x+1 and NJ=3x+5, what value of x makes point N the centroid? K, J M

A: (1).   If in the given triangle N is the centroid then the point N divides the line HL, JK and LM in…

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A: Consider the diagram and divide into four regions as shown below, Now, make a chart and evaluate…

Q: 5. Use sheees to tirel the udure di the s reuolutian: Reçim bameled loy about Hhe axes : Y2x ) メ-0…

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Q: Calculate fol(3z + 4y) ỉ + (5x + 2y) ) · dř where C is the circular path with center (a, b) and…

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Q: Find the moving tri hedral of f R CE) = t, 2E> at %3D

A: This is a problem of differential Geometry.

Q: D

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Q: 5.2 Compute I = e -dz, where C is the boundary of the square centred at the origin, with sides of…

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Q: Suppose the temperature at the point (x, y, z) on the sphere x2 + y2 + ?2 = 16 is T= 800xy2z3 .…

A: Solution is given below:

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A: We need to sketch the surface formed by the function

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A: Given and u=y-x,  v=y+x where D is the trapezoid with vertices (1, 0), (2, 0), (0, 2) and (0, 1).

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Q: Let D be the half-disc in the xy-plane given by x +y <9. Calculate e-(x2+v)dA

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Q: Find Df and Duf to:- XZ on the point (,,1) inthe direction of V = 2i -3j+6 K. %3D soLiin ansi- lo

A: Directional derivative of the function.

Q: Show that the value of ∮C xy2 dx + (x2y + 2x) dy around any square depends only on the area of the…

A:

Q: 8. Describe the motion of a particle with position (x, y): x = 3sin (t),y 4cos (t)

A: To describe the motion of a particle for below curve. x=3sint, y=4cost

Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: r 64 In(y) 49 VY

A: We need to solve definite integral.

Q: 6. What is an equation for the largest ball centred at (0,0, 1) that can fit in the paraboloid z…

A:

Q: Let B be the ball with center at the origin and radius 1 SS S6(x² + y² + z²)dV = 2 %3D B 12 167 24

A:

Q: Find the length of the spiral r = eθ/√2 from θ = 0 to θ = π.

A: Given Differentiating r with respect to thita ,we get

Q: Let B be the ball with center at the origin and radius 1 SSS 4(z² + y² + z²)dV = B

A: Explanation of the answer is as follows

Q: Verify Green's theorem for Q7 [x(x? + y²)dx – y(3x² – y²)]dy Where C is the boundary of the triangle…

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Q: Suppose that the Celsius temperature at the point (x, y, z) on the sphere x2 +y? + 2? = 1 is T =…

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Q: Let B be the ball with center at the origin and radius 1 3(x² + y + z?)dV = %3D B

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Q: 2) Find the length of the spiral: r = 0² from 0 = 0 to 0 = V5?

A:

Q: 1. Sketch carefully the solid enclosed by the three cylinders x + y* – 1,x² + 2² = 1, and y + z² =…

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Q: dx dy, where D is the disk x2 + y² < 9. (x² + y?)3/2 Evaluate

A: The given integral is,

Q: 4. If M is a centroid of triangle WOR and WM = 16, what is WX? %3D W Y M R

A: centroid:- i). It is the intersection point of all the medians of a triangle. ii). It divides the…

Q: (8) Find the point(s) where the line r(t) = ti+(1+t)j+5k intersects the paraboloid z = x² + y?.

A: A detailed solution is given below.

Q: 23. Evaluate II (x² + y² + 2²)“ dV, where B is the ball with center the origin and radius 5.

A:

Q: Find /| F-dS if F(x, y, z) = 2xi+2yj+2zk and S is the part of the paraboloid z = = 8-x?-y? lying…

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Q: Calculate fc((7x + 6y) i + (4 + 7y) j) · dr where C is the circular path with center (a, b) and…

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Q: 1 Evaluate dV, where E lines between the spheres a? + y? + z² = 25 and %3D x² + y? + z2 49 in the…

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Q: 12- Evaluate JJI (x2 + y? + z?)dvwhere B is a ball with center the origin and radius 5. B

A: Given problem is :

Q: Let B be the ball with center at the origin and radius 1 SS f6(n² + y² + z² )dV

A:

Q: 3. Given that g(r, y) = x² + y? subject to ry = 1, show that g(x, y) has no maxi- mum points. (Hint:…

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Q: Use Green's Theorem to calculate (422 – 3yx)dr + (3xy+ y² + 6)dy where C is the circumference (x –…

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Q: 13-Evaluate JSS zdV E where E lies between the spheres x2 + y? + z² = 1and x2 + y? + z? = 4 in the…

A: To find- Evaluate ∭Ez dV where E lies between the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4 in…

Q: 47. Find the distance between the spheres x? + y² + z² = 4 and x? + y? + z? = 4x + 4y + 4z – 11.

A: The equation of the spheres are given:

Q: If z(z, y) = 2x +y?, z = e", and Y = ue", find . %3D %3D Provide your answer below:

A: Put the expression of x &amp; y in z than calculate partial derivative of z with respect to v. When…

Q: 5. USC cyllndrical coordinates. Evaluatc (k4)dv, where E IS the solid that ies betwern the cylinders…

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Q: Evaluate dV, where E lines between the spheres a+ y + z = 9 and E 2 + y2 + z? x2 + y? + z2 = 16 in…

A: The answer is in step 2

Q: 35. Above the cone z = a² + j? and below the sphere x² + y² + z² = 1

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Q: Evaluate +y² 1 Ez² + y² JJJ₁² dV, where E lies between the spheres z² + y² + 2² +2² 16 in the first…

A: Given that,

Q: Let B be the ball with center at the origin and radius 1 SS $3(x² +y? + z²)dV = 16 12m 24m :5

A: