re 2" dr + (x* + 2a²y?)dy where C is the boundary of the region between the circles a² + y? = 1 and æ² + y² = 4 which are lying in the first quadrant of the plane.

Q: Evaluate ſf, (3x + 4y²) dA where R is the region in the upper half - plane bounded by the circles x²…

A: As the areas are circular thus convert the integral into polar form. Now determine the limits of the…

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

A:

Q: Find the surface area of the part of the plane 4+ 2x + 3y that lies inside the cylinder x² + y? 9.

A: The area is given by using double integral

Q: Evaluate y°dz – a°dy, where C'is the boundary of the region between the circles a? + y² = 9 and a +…

A:

Q: Evaluate 4y² dA where D is the region in the upper half – plane bounded by the circles x? + y² = 1…

A:

Q: Evaluate o y'dr - r'dy, where C is the boundary of the region between the circles z + y- 9 and 2 + y…

A: Here, we need to evaluate . By Green's theorem,                ∮Cy3dx-x3dy=∬-3x2-3y2dD=-3∬r2(r) dr…

Q: Compute 3xy dS, where S is the surface of the tetrahedron with sides z = 0, y = 0, x + z = 1, and x…

A:

Q: ? Find the surface area of the plane ( 15x-5z+20y%3D35) if it is in the first octant

A: Here we have to find the surface area of the plane 15x-5z+20y=35 ................... (1) for the…

Q: Which one of the following surfaces is the aph of f(z, y) = y -2 ? 3..

A:

Q: 1. Find the area of the portion of the plane Зх — Зу + 2 — 12 that lies inside the cylinder y? + 22…

A:

Q: 1. Find the area of the part of the surface z = a? + 2y that lies above the triangle with vertices…

A:

Q: Find the surface area of the plane ( 15x- ? 5z+20y=D35) if it is in the first octant

A:

Q: Find the point lying on the intersection of the plane x + y +z = 0 and the sphere x2 + y2 + z? = 25…

A:

Q: Verify Green's theorem for Q7 [x(x? + y²)dx – y(3x² – y²)]dy Where C is the boundary of the triangle…

A:

Q: Consider the portion of the plane x + 2y + 3z = 12 that sits above the circle (x − 2)^2 + (y − 2)^2…

A: The solution are next step

Q: Which of the following represents the area of the surface cut from the plane z = ax + by by the…

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

Q: 10. Find the area of the portion of the sphere x2 + y² + z² = 16 between the planes %3D z=1, z=2.

A:

Q: c) By using Green's theorem, evaluate (In (r) + ry – y²) dx + (e" +x²y - 3ry) dy, where C is the…

A: Green's therorem establishes relationship between double integral and line integral. Double integral…

Q: Verify Stoke's for F = (2x – y)i – yz²j – y²zk where S is the upper half of the sphere x? + y? +z² =…

A: Solution is given below

Q: 17. Evaluate the integral: S, 9x? – 6dx. 18.Determine the center of the circle x² +y² = 25. %3D

A: (17)  Given that I=∫-159x2-6 dx It is known that ∫xndx=xn+1n+1 So, I=9x33-6x-15 =3x3-6x-15 Now we…

Q: 8. Find the surface area of the part of the sphere x2 + y? + 22 : = 4z that lies inside the…

A: The solution is given as

Q: Evaluate fF.ndS, where F-4xi-2y' j+z'k and S is the surface bounding the region x' +y' =4, and the…

A:

Q: Find the projection in the ry plane of the curve of intersection of the surfaces 222 = r² + y? and r…

A: Find your answer below

Q: / (2² + y²) do, where S is the portion of the sphere r? + y? + z² = 9 in the first octant that is…

A:

Q: Find the area of the part of the sphere x2 + y? + z² = 4z that lies inside the paraboloid z = x2 +…

A:

Q: O By using Green's theorem, evaluate $ (In (2²) + æy² – y²) dæ + (ev + x²y – 3ry) dy, where C is the…

A:

Q: Find the area of the band cut from the paraboloid x² + y² – z = 0 by the planes z = 2 and z = 6.…

A: Given equation is, x2+y2 = z Band cut from the given paraboloid by the planes z = 2; z = 6 Area of…

Q: Find the region in the xy-plane in which the following equation is hyperbolic: ((x- y)? - 1]u + 2u,…

A: Given partial differential equation x−y2−1uxx+2uxy+x−y2−1uyy=0 We have to find the region in the…

Q: 10. Using Stoke's theorem, evaluate y dx+ xz dy+x² dz where C is the boundary of the triangle cut…

A:

Q: 3. Evaluate ox?y dx + xy? dy using Green's y 2 Theorem, where C is positively oriented by line y =x…

A:

Q: Evaluate y°dx - r'dy, where C is the boundary of the region between the circles + y = 1 and 7? + y =…

A:

Q: Find the surface area generated by rotating the semi-circle x = (49 – y² be- tween -7 < y < 7 about…

A: To Determine :-  The surface area generated by rotating the semi-circle x = 49 - y2 between -7 ≤y ≤7…

Q: Evaluate ffp(4xy + 1) dA, where R is the region in the upper half-plane bounded by the circles a? +…

A:

Q: Use Green's Theorem to evaluate (1in z – 2y) dz + (x – ycos y) dy, where C is the circle (x – 4)² +…

A: Given: C is the circle x-42+y-42=9. ∮Cxsinx-2ydx+x-ycosydy. To find: Using Green's theorem, value…

Q: S y = 3 – 2 4z – 4 = 3(x +1)?

A: Let us start solving the problem step by step

Q: 56 Consider points P and Q in the x-y plane, with P = (1,0) and Q = (0,1). The line integral %3D Q…

A: Line integrals

Q: 5. USC cyllndrical coordinates. Evaluatc (k4)dv, where E IS the solid that ies betwern the cylinders…

A:

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

A: Sol

Q: Find the area of the part of the sphere x+y +z = 16z that lies inside the paraboloid z =x²+y?.

A:

Q: B is the first quadrant part of the region enclosed by the circles x2 + y? = a? and x2 + y? = b2…

A:

Q: Find the area of the part of the plane x + 4, y + z = 4 that lies in the first octant. 1. A = 12 2.…

A:

Q: Evaluate (xr + y² + =³)? dV, where 2 is the solid hemisphere (half of the Ω sphere) enclosed by x +y…

A:

Q: Find the area of the surface. The part of the plane x + 2y + 3z = 1 that lies inside the cylinder x2…

A:

Q: Evaluate || 4.nds , where A=zi+xj – 3y°zk and S is the surface of the cylinder x² + y =4 included in…

A: Let ∅=x2+y2-4  Then the normal to the surface S is ∇∅=2xi+2yj The unit normal is…

Q: 3. Find the area of the surface cut from the paraboloid x? + y +z = 2 by the plane y = 0.

A: Here we have to find the area of the surface cut from the paraboloid x2+y+z2=2 by the plane y=0.

Q: c) By using Green's theorem, evaluate (In (z²) + ry? – y²) dr + (c" + r*y-3ry) dy, where C is the…

A: We have to evaluate ∮Clnx2+xy2-y2dx+ey+x2y-3xydy where C is the region bounded by the circles…

Q: Find the area of the part of the plane  5x + 3y + z = 15  that lies in the first octant.

A: From the given statement, the plane is 5x + 3y + z = 15 and it is lies in the first octant.

Q: Evaluate the line integral: x dy – y da x² + y² where C is the circle x² + y? = a², a > 0, oriented…

A:

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.