1. Evaluate the triple integral ∭E x dV

Q: 46.) (a) Parameterize S the 1st octant portion of the cylinder y² + z² = 9, 0 <x<4 (b) Use above to…

A: y2+z2  0≤x≤4x=uy=3 cos vz=3 sin vparameterise the surface ass(u,v)=(u   3 cos v   3 sin v)0≤u≤4,…

Q: Consider 6x (1+x²)² 2(x+y) (1+x²)² + 2y ý = Using the function V(x, y) = 12 + y² show that the…

A: What is Lyapunov Stability Theorem: Lyapunov stability theorem says that if a function is positive…

Q: Example 3 a) Evaluate the surface integral 2x²y dS over the surface y² + z² = 1 between x = -1 and x…

A: Note: We are entitled to solve only one question at a time. As the specified one is not mentioned,…

Q: Jse Green's theorem to evaluate line integral | V8 + x3 dx + 9xy dy where C is a triangle with…

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Q: Consider the pendulum x' Y, y' = -vy - sinx, - where 0 is a friction parameter and (x, y) E S¹ x R…

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Q: Evaluate the triple integral ∫∫∫T (x^2) dV where T is the tetrahedron with vertices (0, 0, 0), (1,…

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Q: 14 13 12 11 10 9 8 7 4- 3- 2- Find the volume of the solid S obtained by rotating R about the line y…

A: Here R is the region bounded by y=9x,x=2.5 and the x-axis i.e.(y=0)and the solid…

Q: 1. a) Use the cylindrical coordinates to evaluate (3 + xy)dV, where E is the solid E in the first…

A: Here is the solution of the given problem.

Q: II| (x + y + z) dv, where E is the solid in the first octant that lies under the paraboloid z = 16 -…

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Q: Evaluate the triple integral I/| (x, y, z) dV over the solid E. f(x, y, z) = vx2 + y², E = {(x, y,…

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Q: 19. Evaluate II \xyz| dx dy dz over the solid ellipsoid x² y2 < 1. a? (Hint: Let x = au, y = bv, and…

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Q: 16 The area under the curve 4y? (x2 + 1)3 was rotated about a line located to the left 9 of y – axis…

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Q: O Line integral ) K+Z)dx+Cx-y)dy+ xdz C is ellips de fined by +* =/ and Z=1 aus. 67 Hint:-use the…

A: Given the given line integrals are (1) ∫Cx+zdx+x-ydy+xdz C is an ellipse x24+y29=1 z=1 (2)…

Q: Let E be the solid region in the first octant bounded by the coordinate planes, the cylinder x2 + y2…

A: Let E be the solid region in the first octant bounded by the coordinate planes, the cylinder…

Q: 3. Evalnate. the. fallowing.denble.integrals..by.convecking.bo. Can polar.Caa.r.dinates. ... 2 2…

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Q: Given the curve y = (sin x)² 1. Compute the area bounded @ x = 0 to x = 2π.  a. π sq.units b. 2π…

A: Area of the bounded by the curve : If y=f(x) be an continuous function on a,b , then the Area…

Q: 6) Find the centroid of the region bounded by the curves y = 4x and y = x?. Show how to find the…

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Q: 2. Show that the surface area of the solid of revolution obtained by revolving the curve y = 1/x, x…

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Q: 2 2 5. Find the volume of the solid bounded by the surface f (x, y) = 3x´y´ and the curves y = 2x -…

A: We will have to find the volume of the solid bounded by the surface f(x,y)=3x2y2 and the curves…

Q: SI. f dV here D is the solid region in the first octant between the elliptic cylinder 4x2 + 22 = 16…

A: Find the limits of x , y,z then compute integral

Q: Since the R integration region is the region bounded by the hyperbolas x2 - y2 = 1, x2 - y2 = 9, xy…

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Q: Evaluate the triple integral [ 2³ dV, where E is the solid tetrahedron with vertices (0,0,0),…

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Q: 4) Evaluate , z dV using rectangular coordinates, where Q is the solid in the first octant bounded…

A: In the first octant,  x≥0y≥0z≥0    Thus x, y and z cannot be less than 0

Q: Evaluate the triple integral I = II 3(x2 + y²) dV where D is the region inside the parabolid z = 9 –…

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Q: 2. Using Green's theorem, evaluate | (ev + 4x°y³ – ya) dx + (xe" + 3x*y? – 3x - :- cos y) dy, aD…

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Q: ¹ [ ] [ ²0 x dxdydz where E is the solid limited by Plane E z = 0,z=x+y+5 and by the cylinders x² +…

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Q: 5. On the surface z-3x^2+2y^2 (an elliptic paraboloid), (x,y) changes from (1,1) to (0.99,1.02). Use…

A: Since you have asked multiple questions, we will solve the first question (Question 5) for you. If…

Q: 11. The y-coordinate of the centroid of the solid in part IV is.. A. 0 В. 1 С. 2 D. 3

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Q: The solid S is obtained by rotating the region enclosed by y = f(x) 2 0, a = 5+ 5e6 1- e6 5+ 5e16 1-…

A: The given question is solved below.

Q: (a) Let E be the solid region bounded by the cylinder x + y xy-plane, and below the plane z = 7.…

A: According to the given information, the given integration can be written as:   Part (a):

Q: b)I bounded by y 2 0, z 2 0, x > 4z, and 16 > x + y. Evaluate the following triple integral: Let E…

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Q: (15) Evaluate ///, y dV where D is the solid cut from the cylinder 2² + y? = 1 by the planes z = 0…

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

Q: Evaluate II| 6x(x2 + y2) dv, where E is the solid in the first octant that lies beneath the…

A: Since the integral lies in first octant so 0≤θ≤π/2 and z≥0.  Converting into cylindrical coordinates…

Q: Evaluate the triple integral I = II 3(x2 + y²) dV where D is the region inside the parabolid z = 16…

A: Integration is the process of finding the anti-derivative function of a given function. The area…

Q: 2. Evaluate [(3yz)dV where B is the region in the first octant lying below the plane x + 2y +2: = 4.

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Q: need to find the limits of integration in terms of ρ, φ, and θ.

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Q: No. Date Evaluate fff dv G x² + y² where G is a solid that lies inside cylinder *+(4-2)² =4 obove…

A: we will use cylindrical co ordinate system to solve the given integral

Q: e-(z²+y²+z²) dV. x² + y² + z² where E is the region bounded by the spheres x? + y² + z² 1 and x? +…

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Q: 5. Let E be the solid region above the plane z = 2 and below the sphere ² + y² + z² = 16 (see…

A: Let E be the solid region above the plane z=2 and below the sphere x2+y2+z2=16

Q: (b) Find the surface area of the portion of the paraboloid z-2-x -y above the xy- plane. AMINATION A…

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Q: c, y, z) = x²zi + xz'j+ y ln(x + 1) k, s the surface of the solid bounded by the 2z = 4, y = 3, x =…

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Q: 15. Use the indicated change of variables to evaluate the double integral ||18xy dA where R is the…

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Q: 3. Find the volume of the solid under the plane 3x+ 2y – z = 0 over the shaded region between the…

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Q: 6. † Make a sketch of the region S = {(x, y) : 0 < y <1– x, 0 < x < 1} and express the double…

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Q: (1) Sketch the regions of integration. (a) 7/2 Jo r3m/2 (b) (1r, 0)r drdo 3n/4

A: Consider the equation

Q: 1. Evaluate the line integral I = = [v¢4x° + y*) dx + x(2x° + 3y*) dy] around the ellipse x/a? +…

A: The given line integral is, I=∮y4x2+y2dx+x2x2+3y2dy We know the Green's theorem:…

Q: The volume of the solid generated by the curve y = e¯ª, between x = 2 and x = 7 about the x – axis…

A: Given curve, y=e-x x=2  and   x=7

Q: 1. Evaluate using cylindrical form, I. yz dV where S is the solid in the first octant enclosed by y…

A: Given : y = 0 and y = 1-x2 and z = x To find : Area enclosed by given solid. Solution : The…

Q: The volume of the solid generated by the curve y = e¬ª, between x = 3 and x = 7 about the |3| x –…

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