1. Under the paraboloid = 3z2 + y? and above the region bounded by y = x and r= y² – y

Q: 2) Find the centroid of the region above the X is bounded by the Y axis and the line Y =

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Q: (b) / = xyz dV if S is the region bounded by the cylinders x² + y² = 25, x² + z² = 25 and the first…

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Q: 4/ Find the area of the region enclosed between the curves y= cos 2x, y=0, x=TT/4, x= TT/2 ? ans/…

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Q: x² sin xy 14.5.8. || - dA(x, y), if D is the region bounded by æ² = "4, x² = TY, y² = ;, and D y? =…

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

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Q: 12. In the xy plane the region D is bounded by the lines y = 2x + 6 y = 2r +1, y = -3r +6, and y =…

A: Given: y=2x+6 y=2x+1 y=-3x+6 y=-3x+1

Q: Determine a region of the xy-plane for which the DE (y- x) y' = y+ xwill have a unique solution…

A: Consider a differential equation y'=F(x,y), y(x0)=y0. Then there exist a unique solution to this…

Q: 4. Evaluate S, cos dA where D is the region bounded by D = {(x, y)/ VL < x < v3Ln , 0 < y< x² }.

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Q: ex: Evaluate SS (&++y²)dV over the first octont region boundeol by the cylinders x*ty=/ and xay=4…

A: Convert coordinate system into cylinder coordinate system. Sketch region D and find range of…

Q: 1. Compute  E yz dV , where E is the region above z  0, below z  y, and inside x 2 y 2  4

A: Obtain the limit of integration as follows.

Q: Evaluate (2r+y) dV _ where V is closed region bounded by the z= 4>x and the planes x = 0, y = 0 and…

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Q: EXAMPLE 2 to evaluate the Use the change of variables x = u² - v², y = auv integral SSR y dA, where…

A: The objective is to evaluate the integral using change of variable. x=u2-v2, y=2uvy2=4-4x, y2=4+4x,…

Q: Use Green's Theorem to evaluate the line integral Icov for the given path. (y-x) dx + (2x - y) dy C:…

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Q: Volumes of solids Find the volume of the following solids.

A: The region is bounded by the curves, The region is shown in the graph given below,

Q: (b) y T= x (p) 3. Region bounded by: y = 4 – x² and y = 0. Rotate about: (a) the x-axis (с) у — —1

A: I have solved the first part for next parts repost the question

Q: 17. Region bounded by y = 2x, y = x and x = 2. Rotate about: (a) the y-axis (b) x = 2 (c) the…

A: As per company guidelines we are allowed to solve maximum three sub parts so i am solving first…

Q: 3. Sketch the region and evaluate the iteraterd integral (a²y – ry?)dydx. 2.0 1.5+ 1.0 0.5 0.0 0.0…

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Q: 2 Evaluate 2 dA where D is the region described by 0s ys 55 - x Integral is equal to : Enter an…

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Q: 16. By inspection, a simply-connected and continuous region R A. has no holes and is closed. B. has…

A: Introduction: The application of connected regions is very significant in different fields of…

Q: Evaluate xydxdy over the region bounded by x axis, ordinate x = 2 and the curvex² = 4y %3D OA 3 О В.…

A: The shaded region shown in the figure is the area of integration

Q: Hw.3 3 Find the areas for the following Vegions: Region boundded by y=x y , and Region bounded by…

A: To find the area of the following region :-

Q: 3) Evaluate the double integral Sla dr dy where D is the region bounded by the hyperbolas ry = 2, ry…

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Q: 9. Evaluate 18x dV where E' is the region behind the surface y = 4 - 2² that is in front of the…

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

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Q: 6. Compute the integral zez²+y²+z² x2 + y? + z2 dV(x,y, z) over the region P = {(x, Y, z) ; x² + y²…

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Q: 11. xyz dV, where G is the solid in the first octant that is bounded by the parabolic cylinder z = 2…

A: To evaluate the triple integral

Q: 3. Use Green's Theorem to evaluate (e3x +y²)dx + (sin y – x²)dy where C is the boundary of the…

A: Green's theorem

Q: 4. Evaluate fe cosy dx + (xy – xsiny) dy where C is the boundary of the region lying between the…

A: Given  ∫Ccosydx+(xy-xsiny)dy

Q: 5. Show that, in the region bounded by y = Vx and y = x2 the following double integral 3 (х + у) dA…

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Q: Use Green's Theorem to evaluate the line integral (y – x) dx + (2x – y) dy for the given path. = x²…

A: Green's Theorem: By Green's Theorem we have for a differentiable function f(x,y)=M(x,y)+N(x,y) and a…

Q: set up the double integral for the area of the portion of the paraboloid z=x^2+y^2 that is between…

A: z=x2+y2∂z∂x=2x∂z∂y=2y

Q: 1) Find the centroid of the region that is bounded by the line Y = 3X, the line X = 2 and the X…

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Q: Use Green's Theorem to evaluate the line integral (y - x) dx + (2x - y) dy for the given path. C:…

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Q: Calculate the Pntegrals over the defined region . y=-2x and y=1-2x on paralleberaM clased with Rhes.…

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Q: Evaluate the triple integral. SSLĘ 8x dV x = 2y2 + 2z2 and the plane x = 2. where E is bounded by…

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Q: Question 1. Use a change of variables to compute the integral /I. rydxdy, x3, y = 2x3, x = y³ and…

A: We have to compute the double integral ∬xydydx over the region R which is bounded by the curves…

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

A: Use greens theorem

Q: 2. Evaluate [(3yz)dV where B is the region in the first octant lying below the plane x + 2y +2: = 4.

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Q: The centroid of the region bounded by sin x < y< cosx, and 0<x<7/4 is, TV2-4 b) 1 1 a) 4(42 +1) 4(v2…

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Q: Use Green's Theorem to evaluate the line integral √(x − x) dx + (2x - y) dy (y- for the given path.…

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Q: The region D bounded by y = 0 and y = -1+a² is shown in the following figure. Is this region of Type…

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Q: b.) xyz dv if S is the region bounded by the cylinders x² + y² = 25, x² + z² = 25 and the first…

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Q: 13. Region bounded by: y = 4 – x and y = 0. - Rotate about: (а) the x-axis (c) y = -1 (b) у — 4 (d)…

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Q: 3. Find the centroid of the region bounded by y 3-e, y0, 0, and 2. Since 0, z = 0, and x-2. Since…

A: The given region is bound by three straight line i.e. y = 0, x = 0 and x = 2 and one curve y =…

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

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Q: Use Green's Theorem to evaluate the line integral (y - x) dx + (2x - y) dy for the given path. C:…

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Q: 7. Find the centroid of the region bounded by the graphs of Vx + Vy = va; x = 0; y = 0

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Q: 8) consider the region R in the first quadrant(xz0, y2) bounded by x=2y² X=Y²+1 a) set up, but clo…

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Q: 2. 7 3. Find the centroid of the region bounded by y 3, 3 to the nearest thousandth. nd Since you…

A: To find the centroid of the given region R by integration

Q: 2) Find, without using Pappus' theorem, the centroid of the region above the X axis that is bounded…

A: Given curve is                                                x2+y2=1 this is the circle whose…

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.