d h2 (y) f(x, y) dx dy hị (y) for the volume of the region enclosed by the surfaces z = 2 – x² – y² and z = 2x + 2y. What is ha(0) + f(0 0)?

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: (B) Integrate g(x,y,z)=x+y(x+2z) over the surface area of the region cut from plane x+2y+2z=5 by the…

A: What is Surface Integral: Surface integral is an iterated integration that is performed over a…

Q: S.(2x° +5y )dx+(2x-4y') dy C:boundary of the region lying between the graphs of x² +y² = 16 and x +y…

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Q: f ay) =xy(3-x=y) what is the absolute maximum and Minimum absolite of the funtion R reğion in

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Q: Find [[fw ryz dV if W is the solid region in R³ below the surface z = x² + y² and above the square 0…

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Q: Evaluate p (x² – 2xy)dx + (x²y +3)dy around the boundary of the region defined by y? = 8x and x = 2…

A: To evaluate : ∮x2-2xy dx+x2y+3 dy around the boundary of the region defined by y2=8x and x=2 (a)…

Q: 1. Find Sp(r+2y)dA, where D is the region bounded by the parabolas y = r2, and y=1+2

A: I am going to solve the given problem by using some simple calculus to get the required result.

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: |(x+2y)d4, where D is the region bounded by the parabolas y=2x² D and y =1+x.

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Q: Evaluate E=∭D(x2+y2+z2)−3/2dV where D is the region bounded by the surfaces x2+y2+z2=9 and…

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Q: THE REGION BOUNDED BY 2y = x AND (x-2)² = − 4y + 12 15 REVOLVED ABOUT THE LINE Y = 4. DETERMINE THE…

A: Introduction: The washer method refers to this application of the slicing method. Because the slice…

Q: 1. Find the area of the surface z = k + xy inside the cylinder x² + y² = R² for constant R. k > 0 is…

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Q: I| (12y – 8x) dV where E is the region bounded by y = 10 – 2z and in front of the regic bounded by z…

A: Please refer to the image below.

Q: 3. Find the volume of solid of revolution (in cubic units) when the area of the region bounded by y…

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Q: 8. (x – y) dV, where E is enclosed by the surfaces z = x² – 1, z = 1 – x² , y = 0, and y = 2 E

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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: Suppose that f(z, y) = /4 – a² – y² at which {(r, y) | 2² + y² < 4}. Then the double integral of…

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Q: Evaluate SS x² y4 dx dy D 1 where D is the region bounded by the hyperbolas xy-2, xyalk parabolas y²…

A: We have to Evaluate the integral  ∬Dx2y-4dxdy where D is the region bounded by the hyperbolas xy=2,…

Q: Evaluate the triple integral. (x - y) dv, where E is enclosed by the surfaces z = x² - 1, z = 1 -…

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Q: (iii) Sf xy²e*dA, Dis bounded by y = x,y = 2,x = 0.

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Q: T// Vx² + y² dV, where E is the solid in R3 bounded by the surfaces E z = x² + y² and z = 8 – x² –…

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Q: Q. N. 3 Find the volume of the solid generated from revovling the area of the region bounded by the…

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Q: Evaluate the triple integral x dV where E is the solid bounded by the paraboloid x = 9y² + 9z2 and x…

A: Given the region

Q: valuate the double integral. (x2 + 2y) dA, D is bounded by y = x, y = x³, x 2 0

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Q: Q3 Verify Green's theorem in plane for (x² - 2xy) dx + (x²y + 3)dy where c is the boundary of the…

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Q: 1. Evaluate the double integral r²y dA, where D is the region enclosed by the x-axis and the…

A: We can evaluate the double integral.

Q: Find a. L{y(t)} , b. y(t) in integral form of convolution y"+y'+3y=h(t-4), y(0)=-2, y'(0)=2

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Q: Evaluate the triple integral. (x - y) dv, where E is enclosed by the surfaces z = x² - 1, z = 1 -…

A: First, draw the region E, bounded by z=x2−1,  z=1−x2,  y=0, and y=8. The region is bounded below by…

Q: 2 2 5. Find the volume of the solid bounded by the surface f (x, y) = 3x´y´ and the curves y = 2x -…

A: We will have to find the volume of the solid bounded by the surface f(x,y)=3x2y2 and the curves…

Q: - 9 + 3 + 3y and Find the volume of the solid bounded by the paraboloids z == z = 8 – 3x - 3y

A: To find volume of the solid bounded by the paraboloidsz=-9+3x2+3y2 and z=8-3x2-3y2

Q: Evaluate f S Sp(2+ e(=²+v²+z*)*/²) dV, where B = {(x, y, z) | x² + y² + z² < 9} Describe the region…

A: Consider the provided question, (2) (a)

Q: Calculate an integral ∬R √ (x2 + y2)dA, where R is the region within the upper semicircle of radius…

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Q: Let A be the region bounded by y = 8x 3(1−x) and the x-axis between x= 0 and x= 1. Find the solid…

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Q: - The region bounded by 2y = x and (x − 2)² = −4y + 12 is revolved about the line y = 4. Determine…

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Q: 2ху dx + (x + у) dy Jc C: boundary of the region lying between the graphs of y = 0 and y = 1 – x2

A: Given: ∫C2xy dx+(x+y)dy y=0 and y=1-x2

Q: Evaluate the triple integral ∭E (x+7y)dV where E is bounded by the parabolic cylinder y=2x^2 and the…

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Q: Evaluate 1 dv √x² + G where G is a solid that lies inside the cylinder x² + (y − 2)² = 4, above the…

A: Please see the attachment

Q: The volume of the solid generated by revolving the region between the parabola x =y²+1 and the line…

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Q: x² and t %3D 2) = z where E is the region bounded by the parabolic cylinder y

A: We have to set up the integral according to the given information

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: E is the region behind the surface y = 4 – x² that is in front of the region in the xy-plane bounded…

A: Given integral: ∫∬E18x dV Given that E is the region behind the surface y=4-x2 that is in front of…

Q: Verify that [y- 2x )dx +(2xy +x²)dy = dA where C is the boundary of a region %3D R R defined by y=0,…

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Q: I// sin(x²+y²) dV , where E is the region E = {(x, y, z) : x²+y² < 1, 0 <z< 1}. E

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Q: Evaluate the triple integral (x + 8y) dV where E is bounded by the parabolic cylinder y = 4x² and…

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Q: 1. Find So(r+2y)dA, where D is the region bounded by the parabolas y r, and y=1+

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Q: Evaluate III Vx2 + y² dV E where E is the region that lies inside the cylinder x2 + y2 = 4 and…

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Q: (x - y) dv, where E is enclosed by the surfaces z = x² - 1, z = 1 - x², y = 0, and y = 4

A: Consider the given function fx,y,z=x-y.  The region E is enclosed by the surfaces…

Q: The volume of the solid generated by the curve y = e¬ª, between x = 3 and x = 7 about the |3| x –…

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Q: Evaluate the triple integral (x + 8y) dV where E is bounded by the parabolic cylinder y = 4x² and…

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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.