Evaluate ſ. (3x² + y)dx + 4y²dy, where C is the boundary of the region in the first quadrant by y = x and y = x². Assume that C is traversed in a counterclockwise manner. Evaluate: .directly using Green's theorem

Q: Evaluate y*dz – r*dy, where C is the boundary of the region between the circles r? + y? = 4 and z² +…

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Q: Find the value(s) of c such that the area of the region bounded by the parabolae y=x2-c2 and y=c2-x2…

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Q: Evaluate JJ, xy dA, where D is the region bounded by the curves xy = 2,4y = x² and y = 4. а.

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Q: By making a suitable change of variables, evaluate y SS, = dx dy where A is the region in the first…

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

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

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Q: Find the value(s) of c such that the area of the region 2 bounded by the parabolae y = x² – c² and y…

A: see below the answer

Q: Evaluate |[(x+2y)dA,where D is the region bounded by the parabolas y= 2x D and y=1+x².

A: We can help you with the question. I am not sure how to help you with its attachments.

Q: Use Green's theorem to evaluate O(1-y') dx+(x³ +e" ) dy , where C is the boundary of the region…

A: The integral is ∮C1-y3dx+x3+ey2dy where C is the boundary of the region between the circles x2+y2=4…

Q: Let M be the region bounded by the parabola y = r² and the line y = x + 2. %3D 6. y = r +2 5. 2,4)…

A: Given that, The shaded region bounded by the parabola y=x2 and the line y=x+2 is M.   And the region…

Q: Consider the surface S: z = 4 - x2 ,bounded by y = −5(x2 - 1) and y = 0, just as it is shown in the…

A: We have to find the area of the surface S, and we will use surface integral to do that.

Q: Evaluate ||(x+2y)dA ,where D is the region bounded by the parabolas y = 2x² D and y=1+x².

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Q: Consider the region R bounded by the parabolas y = and y = r - 3.r, as depicted in the graph below.

A: Since you have posted multiple questions but according to guidelines, we will solve the first…

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: (c) By using the Green's theorem, evaluate $(2xy) dx + (4x²y*) dy if C is the boundary of a region…

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Q: For a rectangular region, the boundary conditions are stated as ø(0, y) = 0, 2eta. 29r.b) – 0.…

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Q: Consider the region enclosed by the curves y = a2 +1 and y = 2r + 1:

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Q: Use Green's Theorem to evaluate C |F.nds, where F = (Va+ 3y, 20 + 3y) C is the boundary of the…

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Q: /| (x + 2y)dA D.

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Q: (1- y*)dx + (x° + e*)dy where C is the boundary of the region between the circles x² + y² = 4 and x²…

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Q: Sketch the region enclosed by the given curves. x = 9y2, x = 28 + 2y2 y y 21 35 30 1 25 20 5 10 15…

A: x=9y2,  x=28+2y2 The region is,

Q: Suppose D is the region in the plane bounded by the lines x + y = 0, x +y = 3, x – 2y = 0, and x –…

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Q: Evaluate xy dV where E is bounded by the parabolic cylinders y = x², x = y² E and the planes z = 0…

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Q: 38. Calculate y = x, x = 0, and x = 1. || e* dA, where D is bounded by the lines y =x +1,

A: We have to find ∫∫D ex dA, where D is bounded by the lines y=x+1, y=x, x=0 and x=1

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

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Q: If F = xy?i + yz? j + zx2k, and S is the region bounded by the triangle with vertices (1,0,0),…

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Q: c) Evaluate the line integral (x+2y)dx -2xdy along the following closed path, the circle x+ y= 25,…

A: In parametric form,

Q: Evaluate y*dx – a*dy, where C is the boundary of the region between the circles a? + y² = 16 and %3D…

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Q: Find c > 0 such that the area of the region enclosed by the parabolas y = x - c² and y = c - z is 1

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Q: Evaluate TI Vx + y2 dv, where E is the region that lies inside the cylinder x2 + y? = 1 and between…

A: In this question, we have to evaluate the triple integral by using cylindrical coordinates.

Q: Use Green's Theorem to evaluate nds, where F = (Va+ 3y, 2a + 3y) C is the boundary of the region…

A: Given Vector field is  F=x+3y, 2x+3y Comparing with F=P,Q, we get  P = x+3y            and…

Q: Evaluate ||| Vx2 + y2 dV, where E is the region that lies inside the cylinder x2 + y? = 25 and…

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Q: || x dA, R is bounded by y = x, 2x + y = 6 and the y-axis. R

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Q: Use Green's Theorem to evaluate |(e* +y²) dr + (+r²) dy where C is the boundary of the region…

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Q: D be the region that consists of points (x, y) such that x² + (y - 3)² ≤9. Then, (x² + y²) dA equals…

A: see my solution

Q: Find the value(s) of c such that the area of the region bounded by the parabolae y = x² – c² and y =…

A: Topic = Application of integration

Q: By a suitable change of variables, evaluate ſf, dx dy where D is the region in the first quadrant…

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

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Q: Use Green's Theorem to evaluate nds, where F = (Va+ 6y, 3x + 6y) C is the boundary of the region…

A: Given that        F = < √x + 6y , 3x + 6y > C is the boundary of the region enclosed by y=x-x2…

Q: Evaluate x dV , where E is the E region in the first octant bounded by the sphere x² + y² + z² = 1…

A: given region is in the first octant. using spherical coordinates x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ and…

Q: ...) The figure below shows contours of g(x, y) on the region R, with 9 < x < 15 and 2 < y < 8. 7 6…

A: Consider the provided question, R=x,y| 9⩽x⩽15 and 2⩽y⩽8Using ∆x=∆y=2, break the region R into 9…

Q: Verify that | (y -2x)dx + (2.xy + x² = [] dA where C is the boundary of a region R ây C R defined by…

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Q: : Evaluate ff,(x + 2y)dA, where D is the region in space bounded by y 2x and y = 1+ x2.

A: The given region D is bounded by the curvesy=2x2 and y=1+x2.We will find first the point of…

Q: Evaluate (x + 5y) dV where E is bounded by the parabolic cylinder y = x² and the planes z = 3x, y =…

A: given is  a partial differential equation in dX , dY, dZ we can partially integrate the given…

Q: Evaluate J[(x+2y)dA, where D is the region bounded by the parabolas y=2x D and y=1+x².

A: Solution: Consider the given parabolas are y=2x2 and y = 1+x2 Draw the graph of the given curves and…

Q: If R is the region in the first quadrant bounded by x-axis, y = x 2 and 2x + y = 3, evaluate ZZ R x…

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Q: The value of $ (2xy - x²)dx+ (x+y²)dy, where C is the enclosed curve of the region bounded by y=x²…

A: As we know from the Green's theorem: ∫CMdx+Ndy=∫∫R∂N∂x-∂M∂ydA   I=∮2xy-x2dx+x+y2dy And C: y=x2, y2=x…

Q: Find c > 0 such that the area of the region enclosed by the parabolas y = 390. c and y = c² – a? is…

A: we know that areaA of the region enclosed by given parabola is given by A=2∫-ccc2-x2dx=4∫0cc2-x2dx…

Q: Given fS (x,y)dA = [ S. 5xy dydx. (i) Sketch and shade the region R,

A: see below the answer and figure

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.