As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by y = 8x?, z = 0, x = 1, y = 0 As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the æ-axis. What is the volume of the solid you (the region) create by spinning abou the z-axis? Volume =

Q: The region under the curve y = x² – x between æ = 1 and - x = 3 is revolved about the line x = 3.…

A: Since you have asked multiple question As per our policy, we will solve the first question for you.…

Q: For question 3 – 4, use the Washer Method to determine the volume of the solid obtained by rotating…

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Q: The following problem is similar in spirit to some which were studied by Archimedes and others.…

A: Let Ah be the closed region in the coordinate plane defined by the vertical lines 1 = x and x = h…

Q: sketch the described regions of integration. 0≤y ≤ 8, 1/4 y≤ x ≤ y1/3

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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: The area of the region bounded by the curves 2y2 +2 and z 2y+2 is 2. 3 5 -1 -2 16 17 (A) 3 (B) 3 13…

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Q: Set up an integral to find the area inside the ellipse . Show all steps to demonstrate the trig…

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Q: The area bounded by the parabolas (x − 2)^2 − 9py=0 and y^2=9p(x−2) is to be revolved around x = 2.…

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Q: Rotate the region bounded by x = y² – 6y + 10 and x = 5 about the y-axis

A: Solution :-

Q: The region bounded by x = 1 - y¹, x = 0 is rotate about the line y = -5. The volume of the resulting…

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Q: Find the partial integral. + 3y?) dy t.

A: Given that a  integration and we solve this integration by using general formulas of integration. We…

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

A: Find point of intersection of:  y=xx=y2-y y=y2-yy2-2y=0y=0,2

Q: Sketch the region of integration. 1. dx dy 1 +>

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Q: Rotate the region bounded by y = 6e 2" and y = 6+ 4x - 2x between a = 0 and a = 1 about the line y =…

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Q: Area = 2c 27 dx

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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: Find the volume of the solid formed when the area of the region bounded by the curves x2- 3y=8 and…

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Q: The numerical value of 34 3dA (where D is the region bounded by lines y=0 and x = 1, D and the…

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Q: 5. Find the volume of liquid needed to fill a sphere of radius R to height h (Figure 18). R h…

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Q: 3 The triple integral L (x + 2y – z)dxdydz is equal to

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Q: 4. Illustrate and find the center of mass and moment of inertia with respect to x and y for the…

A: y = f(x) =4-x2   only in First quadrant…

Q: The region D is bounded by the curves y = 5, and y = 5x². Which of the following can be used to…

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Q: The area of the region bounded by the z-axis and the curve = ²t - 1, y = 1-t² equals to: [2(1-1²)e²…

A: Solution

Q: The base of a solid is the region bounded by the graphs of y = 3x,y = 6, and x = 0. The…

A: Given, The base of a solid is the region bounded by the graphs of y = 3x,y = 6, and x = 0. The…

Q: The triangular region Ω has vertices (1, 1), (2, 0), and (2, 3).

A: My=∫12∫-x+22x-1kx2dydx

Q: The volume of the region bounded by x² + y² + z2 =1 is

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Q: 1. Sketch the region corresponding to the double integral rx+2 dy dx x2 and reverse the order of…

A: One of the two main parts of calculus which are concerned with the study of integrals, as opposed to…

Q: 3. The region bounded by y = x³, y = x, x > 0 is rotated about the x – - axis.

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Q: (1) Sketch the regions of integration. (a) 3m/2 (b) (r,0)r drdo 3m/4

A: The integrations  (a). ∫-π2π2∫04f(r,θ)rdrdθ(b). ∫3π43π2∫34f(r,θ)rdrdθ We have to sketch the regions…

Q: 1. Evaluate fL, 2 dv where E lies below the plane z = 2+ 3x + 4y and above the region in the…

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Q: Suppose the following region is spinning rapidly around the x-axis. y = -1/² x + ²/ y 3 y=x 2 3 X -1…

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Q: 2. Find the center of mass for the region bounded by y = 4 - x2 that is in the first quadrant

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Q: 4. у%3D 2х—х2, y = 2x² – 4x

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Q: 3. Find the area of the region that lies inside both curves: r = sin 0,r = cos 0 0.8 0.6 0.4 0.2…

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Q: Use the result of Exercise 37 in Section 6.1 (or the exercise above if your answer is correct) to…

A: Find the volume V of the solid obtained by revolving about the line y=-3 the region between the…

Q: y? = 1? Which of the following integrals represents the length of the upper half of the ellipse 25…

A: Topic = Application of integration  Correct answer = A

Q: Sketch the region R whose area is given by the double integral. 9. 3 6. dy dx

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Q: number 44 and it says use example 4 so i attached that as well

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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: Evaluate 7x dV where E is the region that lies inside the cylinder x2 + y² = 8 E and between the…

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Q: It is possible for a region in the xy-plane to be both type 1 and type 2 at the same time.

A: This is possible.

Q: You are rotating the region bounded by y 2 = x, y = 1, and x = 0 about the line x = 1. Set up the…

A: First, we sketch the region using a graphing calculator  y2=x , y=1 , x=0 The shaded region is…

Q: The numerical value of л за 3dA (where D is the region bounded by lines y=0 and x = 1, D and the…

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Q: 1 Integrate the function f (x, y) = over the region bounded by r-axis, line æ = 7, and the curve y =…

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Q: Sketch the region enclosed by the given curves. y = 4 cos(5x), y = 4 – 4 cos(5x), 0sxS a/5

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Q: O R is the region bounded by y = 2 – x² and y = x² – 6. - %3D

A: 5.)  y=2-x2  and  y=x2-6 red line y=2-x2 blue line y=x2-6

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: If we rotate the region under the curve y = 8 between x = 2 and x = 3 about the x-axis, what answer…

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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.