2. Bounded by the paraboloid = 10 – 322 – 3y? and the plane z = 4 (using polar coordinates)

Q: 1. Prove that fo r*dr + rydy = where C1 is the line segment from (0,0) to (1,0) by using…

A:

Q: |Use Green's Theorem to evaluate [(3x² +y)dx+4xydy where C is the triangular C region with vertices…

A: The solution is given as

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 compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

A:

Q: 1. 3 Consider the parametric surface 7(u, v) = (2uv, u? - v, u² +v²), where u +v < 2. Evaluate the…

A:

Q: Integrate G(x, y, z) = xyz over the surface of the cube cut from the first octant by the planes x =…

A: Here, the surface is in the first octant. Therefore, the integration is

Q: 25. Evaluate F.N dS where F= yl + 2xj –- zk and S is the surface of the plane 2x +ye 6 in the first…

A:

Q: Integrate ƒ(x, y) = sqrt(4 - x2) over the smaller sector cut from the disk x2 + y2 &lt;=4 by the…

A:

Q: 3) Prove thet the xy-plane with constant curva tue k= urve In is o circle of radious

A: We have to prove that a curve in the xy plane with constant curvature k=1r is a circle of radius r

Q: sketch the solid whose spherical coordi- nates (p, 4, 0) satisfy the given inequalities. 0<p<2/cos…

A:

Q: 18. The volume of the solid that lies between the paraboloid z = 2x² + 2y² and the plane z = 6

A:

Q: 14) Find the points on the surface xy- z2 = 1 that are closed to the origin. %3D

A: To find the point on the below surface that is closed to the origin. xy−z2=1

Q: Consider points P and Q in the x-y plane, with P = (1,0) and Q = (0,1). The line integral %3D Q…

A: The line joining P1,0 and Q0,1 is     x-11-0=y-00-1=t(say)⇒x=t+1,y=-t⇒dx=dt,dy=-dt

Q: 1. Find the area of the surface the part of the plane 3x + 2y + z = 6 that lies on the first octant

A:

Q: Find by using polar coordinates in R^2 the volume of the region bounded by the parabolic z=…

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: 1. Find the volume of the solid bounded by the paraboloids z = 6 – x² – y² and z = 2x² + 2y².Use…

A: From the given problem:   z=6-x2-y2  ; z=2x2+2y2   For finding volume we using the triple…

Q: 9. Find the area of the part of the paraboloid z = r² + y² that lies under the plane z = (Hint: Use…

A: Consider the provided question, My objective is to find the area of the part of the paraboloid…

Q: Are there any points on the hyperboloid r2 - y? – 22 = 1 where the tangent plane is parallel to the…

A: Given hyperboloid ;    x2-y2-z2=1  and equation of plne which is parallel to tangent plane  x+y-z=0

Q: (9) Evaluate the surface integral / S, czdS, where o is the part of the plane r+2y+3z = 6 that lies…

A:

Q: Find the volume of a right circular cylinder of radius a = 4.00 with a paraboloid of revolution…

A:

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

A:

Q: 1) Find the volume of the solid that lies between the paraboloid z = 2x + 2y and the plane z = 8.…

A: Dear student, you didn't mention which question is to be solved so according to Bartleby guidelines,…

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

A:

Q: Are there any points on the hyperboloid x 2 − y 2 − z 2 = 1 where the tangent plane is parallel to…

A:

Q: 4. Find the area of the surface S where S is the part of the plane z = 3x +y that lies above the…

A:

Q: 3. Use polar coordinates to find the surface area of the paraboloid z = x² +y² – 5 that lies below…

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: 10. Find the area of the paraboloid x² + y + z² = 2 by the plane y = 0. (Hint: Implicit Surface)

A:

Q: 28. Show that the ellipsoid 2x² + 3y² + z² = 9 and the sphere %3D x? + y? + z? - 6x – 8y – 8z + 24 =…

A:

Q: Calculate the positive value of m so that the x+y+2z=m plane is tangent to the spherical surface…

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: Find a parametrization for the portion of the sphere x2 +y2 + z2 - 16 between the planes z =- 2/2…

A: we have been given a sphere x2+y2+z2=16 between the planes z=-22 and z=2

Q: Calculate ff A.n ds If A = yi + 2xj – zk and S is a 2x + y = 6 plane surface in the first octane cut…

A:

Q: A point moves along the intersection of the elliptic paraboloid z = 6r + 7y and the plane y = 2. At…

A: Given: A point moves along the intersection of the elliptic paraboloid z = 6x2 + 7y2 and the plane y…

Q: (b) Integrate (x, y,z)=(y+z)+1 the paraboloid 4x - y-z -0 by the plane x 1. over the surface cut…

A:

Q: 4. Suppose that VxF= (0,1,-1), S is the plane x + y + z = 1 in the first octant and Cis the boundary…

A: Given: ∇×F=01-1 S is the plane x+y+z=1 in the first octane  and C is the boundary of S. Now, to find…

Q: Find the area of the part of the plane defined by 3z + 2y + z 12 that lies in the first octant.

A: Given equation of plane :  3x+2y+z= 12

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: 6. The part of the surface z = 1 + 3x + 2y² that lies above the triangle with vertices (0, 0), (0,…

A: Given: A surface z=1+3x+2y2 and vertices of triangle (0, 0), (0, 1) and (2, 1). To find: Area of…

Q: Find the extreme values of ƒ(x, y, z) = x2yz + 1 on the intersection of the plane z = 1 with the…

A: consider the following equations  f(x,y,z)=x2yz+1                                  (1)h(x,y,z)=z-1…

Q: Use the polar coordinates to evaluate volumes : 2) Below z = V1- x2 – y2 inside x² + y2 = above the…

A: Using the polar coordinates we have to evaluate the volumebelow z=1-x2-y2 inside x2+y2=14 above xy…

Q: Bounded by the paraboloid z = 10 – 3x? – 3y2 and the plane z = 4 (using polar coordinates)

A:

Q: 12. Let H be the hyperboloid where x² – y² + z² = 6. Find an equation for the tangent plane to H at…

A: As you asked only solution for Q12 so, I answered here only Q12

Q: Evaluate Sſ Ands where A= (x+y*)î – 4xj +4yzk and s is the surface of the plane, 2x+y+2z=8 in the…

A:

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: Find any points on the hyperboloid  x2 − y2 − z2 = 6  where the tangent plane is parallel to the…

A:

Q: 12. Find the distance from the point P(0,0,0) to the plane given by x + y +z= 6.

A:

Q: 1) Find the volume of the solid that lies between the paraboloid z = 2x² + 2y² and the plane Z = 8.

A: The main objective is to find the volume of solid that lies between the paraboloid z=2x2+2y2 and the…

Q: 7. Find the area of the portion of the plane z = 2y that lies directly above the circle x +(y-1)² =1…

A: we have to find area of the portion of plane .and coverting the problem in polar coordinate makes it…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.