2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and has center the origin and radius 1.

Q: If (2) = u(x, y) + iv(x, y) is analytic in a domain D, then using first principles- that %3D dv du…

A:

Q: 15. If the surface S has projection R on the xy-plane, show that F.īdS = S da dy ה.F provided |ñ -…

A: Question: If the surface S has projection R on the xy-plane, show that ∬SF¯·n¯dS=∬RF¯·n¯dxdyn¯·k¯…

Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

Q: Evaluate the surface SS₂ xz ds S is the part of the 2x+2y+z=4 that li- the first octant.

A:

Q: Use Green's Theorem to evaluate the line integral. cx2 - y?) dx + 5y? dy C: x² + y² = 9 (x² .

A:

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

A:

Q: Consider the PDE y Ur + a Uyy + Uz – Uy = r² + y? In what region of the xy-plane is this PDE…

A:

Q: Evaluate : //| cos(x² +y³) dV, where E is the region bounded by the disk x² + y? < 9 E and the plane…

A: From the region of integration which is bounded by x2+y2≤9 and z=1, identify the limits of x, y and…

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: If E is the solid shape bounded by x 2 0,y 20 ,z 2 0 and the plane 2x + 3y +z = 6 evaluate (x + y)…

A: The given question is solved below

Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y°, the cylinder z² + y² = 2, and…

A:

Q: 4. Where does the curve r(t) = ti + t²j+ (3 t2)k intersect the paraboloid z = x2 +y² ?

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5+x +y, the cylinder a2 + y = 1, and the xy…

A: The objective is to evaluate ∫∫∫EezdV, where E is enclosed by the paraboloid z=5+x2+y2, the cylinder…

Q: Let H be the segment of the hyperbola x-y°= | Connecting (1,0) to (2, J3). Evaluate the line,…

A: Given  H be the segment of hyperbola x2-y2=1 connecting the points (1,0) to (2,3)  we have to…

Q: Evaluate /// 203 dV G if G is a solid in the first octant bounded by the plane y + z = 2 and the…

A: Here we have to find range of values for x,y and z and then find triple integral

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

A:

Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

A:

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

A:

Q: Evaluate ez dV where E is enclosed by the paraboloid z = 4 + x? + y², the cylinder x? + y? = 1, and…

A:

Q: (z+ y – 1) cos D

A:

Q: 18. Evaluate § (9 − x² − y²) dV, where H is the solid hemi- sphere x² + y² + z² ≤ 9, z ≥ 0.

A:

Q: Consider the PDE Y Urr + x Uyy + Ux – Uy x² + y? In what region of the xy- plane is this PDE…

A: compare the given equation with Auxx+Buxy+Cuyy+Dux+Euy+Fu=G A=y,B=0,C=x,D=1,E=-1 calculate B2-4AC…

Q: Evaluate /|/. x dV where E is the solid enclosed by the paraboloid x = y²+z² and the plane x = 9

A:

Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder x² + (y − 1)² = 1.…

A: For the given curve we have the following vector valued function: R⇀=2cost. 1+sint,…

Q: • SSS ²³² x² + y² = 2, and the xy plane. Evaluate e² dV where E is enclosed by the paraboloid z = 6…

A:

Q: Verify the Green's theorem for integral x²y dx + xy dy, where C is the boundary described…

A:

Q: Evaluate. where G is a solid that lies in the first octant bounded by x^2 + y^2 = 3z and z = 3.

A:

Q: 2. Evaluate ([.(Vx F). ds, where S is the surface of the paraboloid x2 + y2 + z = 4 above the…

A: We have given F→=(x2+y-4)i+3xyj+(2xz+z2)ksurface is given x2+y2+z=4 and above the xy planediagram of…

Q: Consider the PDE y u + Uyy + u, - Uy = 2 +y? In what region of the xy-plane is this PDE elliptic? xy…

A: The objective is to find in what region of the xy-plane the pde is elliptic.

Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

A:

Q: 11. Evaluate ([(y+z³ )dS where S is the part of the paraboloid x = 4- y – z² that lies in front of…

A:

Q: 3. Evaluate /// 2* dV if G is a solid in the first octant bounded by the plane y + z = 2 and the…

A: triple integral

Q: Q7. Use Green's theorem to evaluate fy² dx + 3xy dy, where c is the boundary of the semi annular…

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: 15. Evaluate SLL:(z+1)dV where E is the solid enclosed by the paraboloid y = x2 + z² and the plane y…

A:

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: 6. Use Green's Theorem to evaluate the line integral fc 2xy dx + (x² + y²) dy, if C is the ellipse…

A: Let R be simply connected region with a piecewise  smooth boundary  'C'. If functions 'M' and 'N'…

Q: Q7. Use Green's theorem to evaluate y² dx + 3xy dy, where c is the boundary o the semi annular…

A:

Q: 1. Compute the average height of the cone z = a> 0. r² + y² above the disk x² + y² ≤ a² for

A:

Q: Evaluate the line integral Sa y dx +z dy, where C is the line segment from (4) (-1, 1, 2) to (2,3,…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder æ? + y² = 5, and…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4+x² + y°, the cylinder x² + y? = 3, and the…

A: Here we have to evaluate ∫∫∫E ez dV , where E is the enclosed by the paraboloid z=4+x2+y2 , the…

Q: A solid G is bounded by the paraboloid z = 4 – r - y², the cylinder 2+ y = 4 and the plane z = 5.…

A:

Q: E) Sketch the solid bounded by the cylinder ** +y = 4 and the planes y+z = 4 and z = 0

A: Here, we need to sketch the solid bounded by the cylinder x2+y2=4 and the planes y+z=4 and z=0.

Q: Example 6: Evaluate ez dz dy dx E Where E is enclosed by the paraboloid Z = 3+ x? + y² , the…

A: The given problem is to evaluate the given triple integral of given function ez for which region E…

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4 + x² + y², the %3D cylinder x? + y² = 3,…

A: We need to evaluate integral.

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

A: Sol

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x + y°, the cylinder + y = 1, and the ry…

A: To Determine: evaluate ∫∫∫eZdV  where E is enclosed by the paraboloid z=5+x2+y2,the cylinder…

Q: 2) Use a change of variables to evaluate , zdV; where Dis bounded by the paraboloid z = 16 – x - 4y?…

A: