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Q: find the center of mass of a thin plate of constant density d covering the given region. The region…

A: Given: y=2x2-4x and y=2x-x2 Graph:

Q: Find the area enclosed by the closed curve obtained by joining the ends of the spiral r = 80,0 < 0 <…

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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: 9–6. Locate the centroid ỹ of the area. 1 m -2 m

A: We need to find the y-coordinate of centroid of the region given. if we have the region bounded by…

Q: panticle of mass m. moves in the one-dimersianal potential U(x) = U. %3D 92 Sketch Ulx).

A: The function is a rule that associates or fixes a number to the particular point x. If , then the…

Q: The standard parametrization of the circle of radius 1 centered at the point (1, 0) in the xy-plane…

A: using the formula to determine surface area revolving to y axis S=2π∫abx(t)(dxdt2+dydt2)dt

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

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Q: Ques tion No. 02: Show that the area of f egion R encleaseod Ly simple closed cuve č is given by a…

A: Area of closed curve, using Green's Theorem: A=∯Sdxdy   =12∯S2dxdy   =12∯S∂x∂x+∂y∂ydxdy…

Q: The velocities (in mi/hr) of an automobile moving along a straight highway over a 2-hr period are…

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Q: : Prove that three dimentional volume element "dx dy dz " is not invariant bu ment is invariant…

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Q: Set up a triple integral in spherical coordinates to c

A: Given, the solid bounded above by the cone x2+y2=z2and bounded below by the sphere…

Q: with a 200 mm q

A: answer is in next step

Q: 52. Show that a spherical cap of height h and radius R (Figure 19) has surface area 2n Rh. Rogawski…

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Q: Consider the area bounded by the points P, (6,6), P2(8,4), and P3(12,4) Find the volume if it is…

A: Given area is a triangle with vertices 6,6, 8,4 and 12,4. And a line x=-3. The region with the line…

Q: A new spherical bacterial particle of radius 2.5 x 10-10 meters was discovered in the fossil.…

A: To find the diameter

Q: A thin plate of constant density covers the region bounded by the ellipse x2/a2 + y2/b2 = 1, a…

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Q: An artist has decided to finish their piece of artwork by balancing it on a fulcrum and putting it…

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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: 14. (a) Give a parametrization for the intersection of y = x² and z = e". (b) Set up an integral…

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Q: Evaluate the integral xyzdV where E lies between the spheres with radius 2 and 4 and above the cone…

A: Using spherical coordinates,

Q: Find the work required to pump the water out of a tank shaped like an upside down pyramid with…

A: We are asked to find the work done to pump the water out of a tank that is shaped like an…

Q: Present a problem with solution solving the centroid of a solid of revolution using the circular…

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Q: Solve the integral. + yz )do; o: part of the plane x + y +== 6 that lies in the first octanct.

A: Solution:

Q: Set up the integral for the Surface Area of the curve witn given rotation: 1 from ( 2) to (1, 1)…

A: Given curve is x=1/y2 IInterval is (1/4,2 ) to (1,1)

Q: . Calculate the flux integral fF.n ds, where C is the positively oriented circle of radius 1…

A: Given function is, Fx,y=x,2y The formula for calculating the flux of the integral ∮F·nds is,…

Q: Consider the parametrization y: [0,47] → R, (t) = (t cos t, t sin t), of a counter-clockwise spiral…

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Q: 2. Use integration in cylindrical coordinates in order to compute the vol- ume of:

A: Given,       U=x, y, z: 0≤x2+y2≤1, 0≤z≤5-x-y

Q: Give an example of a plate such that its center of mass does not occur at any point on the plate.

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Q: The velocities (in mi/hr) of an automobile moving along a straight highway over a 2-hr period are…

A: The midpoint riemann sum to approximate the area of the function fx in the interval a,b with n…

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: Calculate the flux integral (9i+95+9k)· dà where S is a disk of radius 8 in the plane x +y + z = 1,…

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Q: a. Let S be the cylinder with base a circle of radius 2 in the xy-plane centered at (2,3,0) and with…

A: S is the cylinder with base a circle of radius 2 in the xy-plane centered at 2,3,0 and height 3…

Q: Find the area enclosed by the closed curve obtained by joining the ends of the spiral r = 40,0 ≤ 0 ≤…

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Q: Which of the following is equivalent to the surface area of rotation of the function y on the…

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Q: 2/ Find the line integrals of the ellipse+3= ellipse by x = acos(t) ,y= bsin(t) 1 , parameterize the…

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Q: 4) Write the integral to calculate the outward flux through F = (-yx2, xy,-z) across the sphere with…

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Q: The velocities (in mi/hr) of an automobile moving along a straight highway over a 2-hr period are…

A: a) From the given images, the option c) is perfectly matches the ordered pairs. b) From the given…

Q: Q1/ Find the volume of the solid generated by revolving the shaded region about the x- axis for the…

A: To find:- Volume of solid  generated by revolving the shaded region about x axis by disc method…

Q: İki noktalı Gauss kareleme yöntemini kullanarak 2 8x? -dx integraline yaklaşık değer bulunuz. 1+16x4…

A: We have to Evaluate the integral ∫-12128x21+16x4dx by using two-point Gaussian quadrature rule. As…

Q: 12. Find the surface area of the solid obtained by rotating about the x-axis for 1sxs3.

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Q: Use three-particle interaction to show that mass is an additive.

A: We have to use three-particle interaction to show that mass is an additive.

Q: The arc y=2 sqrt(x) from x∈[4,16] is rotated about the x axis. Find the exact surface area…

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Q: Verify the Green's theorem for the following integrol 3, Sinydx + + (x - cosyldy where curve C is…

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Q: (a) Find that the area of the plane region lying to the right of the parabola x = and to the left of…

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Q: Compute the area A under the parametrized curve c(t) = (e',5t) for 0 <t < 1. %3D (Give an exact…

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Q: xplain why the annulus {z: 1 |Z| 2} is not simply connected.

A: By using Cauchy's theorem G is not simply connected.

Q: Consider the area bounded by the points P, (6,6), P2(8,4), P,(12,4) and P,(14,8) a. Find the volume…

A: (a) The area curved is, A=∫6122πrdr=πr2612=π(6)(18)=108π The volume is, V=216π

Q: The velocities (in mi/hr) of an automobile moving along a straight highway over a 2-hr period are…

A: We will plot points and join them to find graph For mid point rule 1st we need to find intervals and…

Q: The velocities (in mi/hr) of an automobile moving along a straight highway over a 2-hr period are…

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Q: Find the area of the triangle with vertices (0,0), (2, 0) and (2,2) by computing a double integral…

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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.