(15) Let D be the half-disk x² + y? < 9, y > 0. Compute / | (a + y) dA D

Q: - || e dA, where D is the open disk D = {(x, y) : x² + y² < 4}.

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Q: Evaluate the integra ∮C -y3 dy + x3 dx for any closed path C.

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Q: 6. Evaluate (7xi - zk) · ndA over the sphere S : x² + y² + z² = 4 (u) by the Divergence Theorem, (b)…

A: The divergence theorem states that  ∬SF·ndA=∭V∇·FdV Where, F is the vector field, V is the given…

Q: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

A:

Q: 2. Let D be the half-disc in the xy-plane given by x +y <9. 2. Calculate e-?+v}dA Select one: a.…

A: Let  D  be the half-disc  in the xy planegiven  by  x2+y2&lt;9calculate ∫∫D  e-x2+y2  dA

Q: 7. Derive the orthogonal trajectories of the family of hyperbolas xy = c. %3D

A: The solution are next step

Q: parametrizes the portion of the sphere x² + y² + z² = 9 that is above the xy-plane and below the…

A:

Q: An equation of the tangent plane to the surface z = 4xy² + 3x²y² at the point (1, 1, 7) is: Select…

A: z= 4xy2+3x2y2F(x,y,z) =…

Q: 28. Show that the circulation of F(x, y, z) = (x², y², z(x² + y²)) around any curve C on the surface…

A: Given figure Stoke's Theorem Here take single integral over closed path C and double integral over…

Q: the surface z = &x + 4y th=

A:

Q: 39. Find all points on the portion of the plane x+y+z= 5 in the first octant at which f(x, y, z) =…

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Q: -JIL e √x² + y² dv, where E is the region that lies inside the cylinder x² + y² = 4 and between the…

A:

Q: 10. Find all the planar and hyperbolic points on the surface o(u, v) = u-3uv².

A: A point on a surface is said to be a hyperbolic point if the Gaussian curvature at p,i.e., κp&lt;0.…

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: f(x, y) = x - y“; R is the disk x +y < 1.

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Q: Find the area of the plane xtay-zt 10=0 that lies above the disk xy-plare

A: Given that, Area of plane is x+2y-z+10=0 It lies above the disk {(x,y)/x2+y2≤4} in x,y plane. The…

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

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Q: Let B be the ball with center at the origin and radius 1 SS [3(x² + y² + z²)dV =

A:

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

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Q: 2. Let W be the union of the first and third quadrants in the xy- plane. That is, let W = xy 0 a. If…

A: According to our company gardening can answer only one question. If a specific questions are asked…

Q: Find vohume fome above ZVy-xLy, below Z0(xy plane) inside xy/ 3.

A:

Q: 3. Find an equation of the tangent plane to the paraboloid defined by z= 5x +4y? at the point…

A: The given equation is; z=5x2+4y2 at the point (-1,1,9).

Q: (a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

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Q: Use the Divergence Theorem to evaluate (6x + 7y + z2) dS where S is the sphere x2 + y2 + z? = 1.

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Q: The tangent plane to the elliptic paraboloid z = 2x2 + y² at the point (1, 1, 3) is %3D 1) 8x + 6y +…

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Q: evaluate $ A. dš over the sphere R = 3 and show whether Divergence theorem holds or not. %3D

A: Calculate surface integral over R=3 or dr=0 ∮A→×ds→=∮R cos2θ,π,0×r2sinθdθdϕ,rdϕdr,rsinθdrdθ…

Q: 1. The part of the sphere x? + y² + z² = a² that lies within the cylinder x² + y² : = ax and above…

A: To Determine :-  The part of the sphere x2 + y2 + z2 = a2 that lies within the cylinder x2 + y2 = ax…

Q: 4. Use Green's Theorem of the Plane to evaluate the following closed path integral. xy2 dx+ 2x2y dy…

A: According to green’s theorem closed path integral over the curve C is calculated by,

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: Green’s Theorem for line integrals Use either form of Green’sTheorem to evaluate the following line…

A:

Q: Evaluate || F-n dS where S is the cylinder (including the disk on the bottom) x² + y² =1 for 0<z<1…

A: First, we plot the given surfaces.Cylinder : x2+y2=1; 0≤z≤1Paraboloid : 3x2+3y2+z = 4; 1≤z≤4.

Q: SSx x dA; R is the portion of the disk x +y s 16 in the second quadrart

A: According to the given information, it is required to find the value of the given integral.

Q: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.

A: we have to find the volume

Q: JJs where S is the sphere x² + y2 + z2 = 1.

A: Given S is the sphere                         x2+y2+z2=1 To evaluate:             ∫∫S3x+3y+z2dS…

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

A:

Q: Find the surfaces on which the function u = 2 (x2 + y²)/z is constar

A: Since u is a constant so, x2+y2z=4a , where a is an arbitrary constant.

Q: O 9/ Compute I - [.lryz) , uhere s is the part g the plane y +x that lies inside the half-cylinder…

A: We know that, dS=1+∂z∂x2+∂z∂y2dydx where S is the part of plane z=y+x. ∂z∂x=1∂z∂y=1 dS=1+(1)2+(1)2…

Q: Find the tangent plane to the elliptic paraboloid z = x² – 2y at the point (1,1,1). - O A. z = 2x –…

A: Given that: z=x2-2y2      and point is:   (1, 1, 1)

Q: Evaluate [[[ =*dxcydz over the sphere x +y² +=² =1.

A: to find:- to evaluate the given integral.

Q: find the flux of f=zk across the portion of the sphere x2+y2+z2=a2 in the first octant in the…

A: Given the field, F=zk The surface S is the portion of the sphere x2+y2+z2=a2 in the first octant in…

Q: Use the Divergence Theorem to evaluate (4x + 7y + z²) dS where S is the sphere x2 + y2 + z² = 1.

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Q: (1) Find the area of the curved surface of the upper half of the sphere x2 + y² + z² = a²(a > 0) on…

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Q: 7. D is bounded by the surface (a2 + y² + z²)² = a²(x² + y? – z²). %3D

A: Sol

Q: 11. Evaluate ([(y² +z? )dS where S is the part of the paraboloid x=4-y -? that lies in front of the…

A: We will evaluate the given integral.

Q: SSx x dA; R is the portion of the disk x + ys 16 in the second quadrart

A: I use polar conversion method

Q: se the Divergence Theorem to evaluate (8x + 12y + z2) ds Is here S is the sphere x2 + y2 + z² = 1.

A: To compute the below integral using divergence theorem.

Q: Find a parametrization for the intersection of the surfaces: z = x2 + y2 and  x2 + 4y2 = 1

A: Find the parametric representation for the intersection of the given surfaces.

Q: Solve the BVP for the disk (exterior Dirchlet problem) v'u = 0, 1< r <o∞ u (1,0) = sin 0 + cos 20

A: The given function is ∇2u=0 The given condition is u(1,θ)=sinθ+cos 2θ We have to find the general…

Q: *6. Let f: R' →R be given by =+4x'z-6xyz+4y-3x'y. Is M=f-((0}) a smooth surface (2-dimensional…

A: Given, fxyz=z2+4x3z-6xyz+4y3-3x2y2 Now, writing the given function in matrix form.…