and C be the boundary curve of the region R= {(x,y)ER²:x²+ s1.y20) oriented counter-clockwise, where ß is a positive real number. If the Let circulation of F along C equals 33, what is the value of B?. B=

Q: Compute volume under the surface x+ 2y2 above the region described by 0sxs1 and 0sysx?, shown below.…

A: Given that surface is x+2y2 and region is described by 0≤x≤1 and 0≤y≤x2. It is required to find the…

Q: Let R be the region bounded by the curve y = r and the line y = 2x. Use inequal- ities to describe R…

A: In this question we have to describe the area of region by using vertical cross section

Q: B is the region bounded by the following graphs: 1 A1: ]y – 1| = ;x A2: (y – 4)² = –4(x – 6) %3D…

A: According to the question,Given curve,A1 : y-1 = 12x,A2 : y-42 = -4x-6

Q: Use Green's Theorem to evaluate the line integral. (x – 2y) dx + (x + y) dy C: boundary of the…

A:

Q: A region R is bounded by the graphs of  y=0.8 (x)^1/2 and y=0.8x.  Find the moments Mx and My of R…

A: Given,

Q: Use Green’s Theorem to evaluate the line integral  (x^2 − 2xy) dx + (x^2 y + 3) dy where C is the…

A: The given problem is to evaluate the given integral in the contour using the green's theorem in the…

Q: Consider the double integral //(y - x) dæ dy I = R where the finite region R is bounded by the lines…

A:

Q: D Evaluate xy dady where D is Hie region bounded x=0, y=0, and x+y= ! by

A:

Q: Let D be a region bounded by the curve x= Jy and the lines y+x= 2 and y=0. Then, which of the…

A:

Q: Suppose D is the region bounded between the z-axis, the curve y = e", and 0 <I<1. Compute the…

A: We need to compute the integral: ∫∫D2ydA Given that, y=ex and 0≤x≤1.

Q: Evaluate the double integral. / 4x2sin y²dA; R is the region that is bounded by y = x³, y = -x³, and…

A:

Q: Verify Green's Theorem in the plane for F = xyî + x²ĵ and where C is the boundary of he region…

A:

Q: Let R be the shaded region defined and bounded by the graphs of C:y = v25 – x², Cz:y =- 5, and C3: y…

A: Solution is given below:

Q: R is the region bounded by y = 4 – x² and the x-axis and it is revolved about y = 4. Sot un the…

A:

Q: Integrate f(x, y, z) = x over the region in the first octant (x 2 0, y 2 0, z 2 0) above z = y2 and…

A: Given that,

Q: Compute volume under the surface x+ 2y2 above the region described by 0sxs1 and 0sysx², shown below.…

A: Calculate the volume: V=∫x=01∫y=0x2∫z=0x+2y2dzdydx   =∫x=01∫y=0x2x+2y2dydx   =∫x=01x3+23x23dx…

Q: (24) x+1 = y°, x= y,y = 0, y = 1

A:

Q: A solid figure is created by rotating the region R around the x-axis. R is bounded by the curve y =…

A:

Q: Sketch the region R whose area is given by the iterated integral. 16 – y2 dx dy 3 2- 2 6 8. 10 12 14…

A: option B

Q: Use Green's Theorem to evaluate the line integral. dx + ху dy C: boundary of the region lying…

A:

Q: Let R be the “triangular” region in the firstquadrant that is bounded above by the line y = 1, below…

A: We first sketch the region using a graphing calculator  The shaded region will rotate about the red…

Q: Use Green's Theorem to evaluate the line integral. | y2 dx + xy dy C: boundary of the region lying…

A:

Q: Let R be the region bounded by the curve y = x and the line y = 2x. Use inequal- ities to describe R…

A: To describe: The region R using inequalities in terms of its vertical cross-sections. Calculation:…

Q: SI xyz dW, B is region bounded by two cylinders, wwwm w В x^2 + y^2 = 16, x^2 + z^2 = 16, and the…

A: Consider the given volume of the two intersecting cylinders given by  x2+y2=16 andx2+z2=16  as shown…

Q: Verify Green's theorem in plane for (x* - 2xy)dx + (x°y+ 3)dy where c is the boundary of the region…

A:

Q: Use Green's Theorem to evaluate the line integral. by² dx + xy c C: boundary of the region lying…

A: The objective of the question is evaluate the definite integral using Green's Theorem.

Q: consider the region in the xy plane bounded above by the curves y=x9/2 and y=x

A:

Q: Consider the region bounded by the graphs of y = 3x - 1 and x = y²-3, illustrated at the right.…

A: (a) To find the center of mass of region where the region is given by following curves. y=3x-1 and…

Q: As the region E is the region bounded by y=x2 , x=y2 , z=0 , z=x+y,  Calculate the triple integral.

A: It is volume integral

Q: Set up a double integral that gives the area of the surface of the graph of f over the region R.…

A: To find the area of surface of graph f

Q: Calculate F. dr. where F(r, y) = (32 + 3y, y? – 7x) and C is the positively oriented boundary curve…

A: Green's theorem relates a line integral around a simple closed curve C to a double integral over the…

Q: Use Green's Theorem to evaluate the line integral. y2 dx + xy dy C: boundary of the region lying…

A: Given problem:-

Q: Find the areas of the regions 1. Bounded by the spiral r = 0 for 0 < 0 < T r = 0 2'2 х (п, т)

A:

Q: Consider calculating the size of the region N:= {(x, y) E R² : y > v3x², æ² + y² < 4} , with the 2d…

A:

Q: While evaluating the area of the region bounded by two curves Vy = z and y- 11z + 24 = 0, the values…

A: Given curves are: y=x   ............................(1)&amp; y-11x+24=0  ..............(2)

Q: Evaluate the moment My and My of the region bounded by y = 2x + 1, x + y = 7 and x = 8.

A: evaluate the moment Mx and My of the region bounded by y=2x+1 , x+y=7 , x=8

Q: Compute volume under the surface x + 2y2 above the region described by 0<x<1 and 0 sysx2, shown…

A: See the attachment

Q: Use Green's theorem to compute f. y³dx – x°dy where C is the boundary of the region between two…

A:

Q: Sketch the region enclosed by the given curves. x = 9y2, x = 4 + 5y2 У y 1.0- 0.5 6. -0.5 o-1.0-…

A: Given curves are

Q: The volume V of the solid formed by revolving the region bounded by the graphs of y=VE-2, y-0and…

A: Let us first find the volume V of the solid formed by revolving the  region bounded by the graphs of…

Q: Evaluate 2x dV Over the region E bounded by the parabolic cylinders y = x2 and x = y² and the planes…

A: Consider the parabolic cylinders y=x2 and x=y2 and the planes z=0 and z=x+y

Q: 1- Integrate The furction F(X,y= over The region bounded byrj=X7y=2x9 x =1/X=?

A: According to our company guideline if multiple questions are posted we must do only first. You can…

Q: Use Green's Theorem to evaluate the line integral Jo for the given path. C: boundary of the region…

A:

Q: Use Green's Theorem to evaluate the line integral. S y² dx + xy dy C: boundary of the region lying…

A:

Q: Q3. The region bounded by the curve y x² and the line y 3x, is revolved about the line x = -3. Find…

A: Here i use cylindrical shell method to find out volume of rotation

Q: Let Q be the region bounded by the graphs of y=x and y=(x^2). Find the volume of the solid of…

A: Given  y=x and y=x2  and Q is the region bounded by these two curves We need to find out the volume…

Q: 4. Compute the value of the double integral fSRe-xydxdy, where R is the region bounded by the lines…

A:

Q: The base of S is the part of the region bounded between y = -4 and y = 4 - that lies in the second…

A: S is the part of region bounded between y=x24-4 and y=4-x24 that lies in the second and third…

Q: Evaluate the integral (y-x)drdy over the region D in the ry-plane bounded by the straight 1 lines y…

A: We have to find the value of the double integral over the region bounded by the given lines. It is…

Q: Evaluate the integral. z dx dy da W is the region bounded by the planesx0, y 0,0, z-1, and the…

A: Evaluation of the triple integrals, uy conversion of cylindrical coordinates

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.