Calculate | [(a* – 3y) dæ + (2y³ + 4x) dy] wherein y(t) = (3cos(t), 3sin(t)), 0

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), E (0,4,0), and (0,0,2)

A: The vertices of the tetrahedron are (0,0,0),(1,0,0),(0,4,0),(0,0,2).  x varies from x:0→1y varies…

Q: Consider AABC with its centroid at point G and median at BH. What is the length of GH if BG = 3x + 4…

A: Given Data: The length of BG is: BG=3x+4 The length of GH is: GH=x+10 The ∆ABC with its centroid at…

Q: 1 The domain of the function f(x,y)= + In(4- 4x2 - y?) is the region inside the x²+4y² – 4 ellipse…

A:

Q: Use the method of Lagrange multiplier to determine the maximum and minimum values of f(x, y) = x* +…

A: Local Maximum and Minimum by using lagranges Multiplayer.

Q: Let f(x, y)=/2x² +4y² + 18 – x. The domain of c for which the level curve f(x, y)=c is an ellipse is…

A: Given fx,y=2x2+4y2+18-x. We need to find the values of c for which the level curve fx,y=c is an…

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (5,0,0), (0,2,0), and (0,0,4)

A: # Given integral is      ∫∫E∫ zdv

Q: Let C be the boundaries of the ellipse 49x2 + 9y? = 49 taken in the counterclockwise direction. A…

A:

Q: Example 8. Evaluate , A.n ds where A=(x+y)i-2j+2yzk and S is the surface of the plane 2x + y + 2z =…

A:

Q: Let f(x, y)=V2x2+4y2+8-x. The domain of c for which the level curve f(x, y)=c is an ellipse is O…

A: Equation of ellipse : (x-h)2a2+(y-k)2b2=1 Where (h,k) is coordinate of centre of ellipse

Q: Find c > 0 c > 0 such that the area of the region enclosed by the parabolas y = x 2 − c 2 y = x 2 -…

A: To find the value of c given area between the curves.

Q: Let a, b be positive real number, F = (esinx + y) i+ (3x – 4(tan y) )j, and C be the boundary of the…

A:

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

A: To Plot: Graph The Surface 1=y2+z2-x24 by plotting at least three cross section in xy and yz plane

Q: prove that the system of parabolas y2=4a(x+a) is self orthogonal

A: We show that the family contain both members with slope m and slope -1/m. Hence the family will be…

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

A:

Q: 4) Consider the images of the hyperbolas x² – y² = ±c² under the mapping f(z) = z², where c > 0.…

A: Given the hyperbolas x2-y2=±c2, c>0. We need to consider the images of these curves under the…

Q: Below the paraboloid z = 18 – 2x² – 2y² and above the xy-plane -

A: The region D lies below the paraboloids z=18-2x2-2y2 and above the xy-plane.

Q: Let C be the portion of the ellipse 4x² + (z – 1)² = 16 on the xz-plane with endpoints at (2,0, 1)…

A: Given : C is the portion of the ellipse 4x2+z-12=16 on the xz-plane with endpoints at 2, 0, 1 and 0,…

Q: Let f(x, y) =V 2x² +4y² +8 - x. The domain of c for which the level curve f(x, y) = c is an ellipse…

A: We have to solve given problem:

Q: Let C be the boundaries of the ellipse +=1 taken in the 9 16 counterclockwise direction. A…

A: We have to check

Q: Find the orthogonal trajectories of the family of curves: Ellipses with centers at (0,0) and two…

A: Given ellipse with centers at (0,0) and two vertices (1,0) and (-1,0).

Q: prove that the system of parabolas y² = 4a(x + a) is self orthogonal

A:

Q: C is the intersection curve between the paraboloid z=x²+y² and

A: Given that C is the intersection curve between the paraboloid z=x2+y2 and the cylinder x2+y2=1.

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), and (0,0,4)

A: x varies from 0 to 1& y varies from 0 to 2 and z varies from 0 to 4.

Q: Use Greens Theorem to evaluate: §. (2x + ev´)dy - (4y2 + e*")dx where C is the boundary of the…

A:

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,3,0), and (0,0,5)

A:

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (4,0,0), (0,3,0), and (0,0,2)

A: we have to find triple integral

Q: Use e- 45° to write 2x + 2.xy + Sy² + óx- 4y = 40 in the x'y'-plane. Then identify %3D the conic.…

A: The following question has been completely solved and explained below

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: Let F (tanx + y)ĩ + (x + secy)j and C be the boundaries of the ellipse 9y=9 in the counterclockwise…

A: Given - F→ = tanx + yi^ + x + secyj^ and C be the boundaries of the ellipse x2 + 9y2 = 9 in the…

Q: Find the vertex of the paraboloid z=x2+4y2−xy+6x−5y+1

A: The paraboloid is of two type, one is elliptic and another is hyperbolic ,  By suitable choice of…

Q: Find the maximum and minimum values of f(x, y, z) = x - z on the ellipsoid x² + 2y? + 22 = 1.

A: maximum=? minimum=?

Q: b) a dr+(x+yz) dy+(ry- VE) dz, C is the boundary of the part of the plane 3r+2y+z in the first…

A:

Q: Let o(r, y, 2) = x2 + (y/2)2 + 22. Find the point on the ellipsoid o = 1 closest to (1, 1, -1).

A:

Q: Let f(x, y)=V2x2+4y2+8-x. The domain of c for which the level curve f(x, y)= c is an ellipse is 0…

A:

Q: 3.4.6 Use Green's theorem in a plane to evaluate of 10x9²³- [(xy²-y) dx + (x + y²) dy] C as a double…

A:

Q: Find the area of the portion of the paraboloid x = 4 - y2 - z2 that lies above the ring 1<=y2 +…

A: Given surface is  The required surface area is given by where

Q: show that a=(2x^2+8xy^2z)i+(3x^3y-3xy)j-(4y^2z^2+2x^3z)k is not a solenoidal but b=(xyz^2.A) is…

A: Given:A=2x2+8xy2zi+3x3y-3xyj-4y2z2+2x3zk To check solenoid we will check ∇.A=0 or not…

Q: Is it true or false that by using a suitable translation x = x' + xo, Y = y' + Yo, it is possible to…

A: Transforming the conic equation

Q: 5) Let R(x, y) = 3x² + y². Find the minimum value of R on the hyperbola xy 2.

A:

Q: D is the region enclosed by the parabolas z = y²-9 and 2 - y² +9.

A: Let, I=∫∫Dx+y+2dA x=y2-9 and x=-y2+9 Power rule for the integration: ∫xndx=xn+1n+1+C ∫cdx=c∫x0dx=cx

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

A:

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 the absolute maximum and absolute minimum of the function f (x, y, z) = x³ + 2y along the curve…

A:

Q: Find the average haight of the Paraboloid z- xry* over the square osx <2, osd<2

A: We need to find the average height of paraboloid

Q: Find the points on the surface z²2=xy+1 that are closest to the origin using LaGrange multipliers

A:

Q: a) Evaluate //(-ri-yj+2: k) n dS, where a is the portion of the paraboloid : = 3r - 3y between the…

A:

Q: Verify Green's Theorem in the plane for F = (x² + y²)î + (x² – y²)f in the anti-clockwise direction…

A: