(4,3) Co,o)

Q: Use differential to approximate the change in the volume of a sphere when its radius changes from 2…

A: Given data:   The initial value of the radius is r0=2 units. The final value of the radius is rf=2.1…

Q: Use integration and trigonometric substitution to find the area enclosed by the ellipse: X^2 /9 +y^2…

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Q: "ind the centroid of the quarter circle x + y <7, y 2 |x| assuming the density ő(x, y) = 1. Use…

A: We need to find mass(M) and moment of inertia about x and y axis of the lamina  Mass of lamina…

Q: Determine the volume charge density, in pC/m3, at (3, -8, -2) if the electric field present in the…

A: In the given question we have to find the evolume charge density at the point(3,-8,-2) in the region…

Q: The linear density of a rod 3m long at a point x meters from one end is ke-3x kilograms per meter.…

A: Solution is given below:

Q: The lateral area L of any prism whose altitude measures h and whose base has perimeter P is given by…

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Q: dv represent? dt If V represents the volume of an object, what does

A: Given question is "V represents the volume of an object, what does dV/dt represent?:

Q: Find the total mass of the given rod and the center of mass The length of a rod is 6m and the…

A: Given that, the length of the rod is 6 m and linear density of the rod is 2x+3kgm. Find the total…

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

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Q: Q/ Find the Surface area generated by revolving y = 1 - 2² around the y-axis, ?0 <x <1 where

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Q: Consider a thin wire of mass m and length L going from x = a to x = b. Let the density of the wire…

A: Given that a thin wire of mass m and length L going from x=a to x=b. The density of the wire is…

Q: 3. Find the surface area of the object obtained by rotating y = 4 + 3x?, 1<x<2 about the y-axis.

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Q: Use integration and trigonometric substitution to find the area enclosed by the ellipse:…

A: Consider the given ellipse  x225+y281=1 Here, the objective is to calculate the area enclosed by the…

Q: Consider a thin metal plate that covers the triangular region shown below. If the density in the…

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Q: Suppose that a thin metal wire in the shape of a semicircle occupies the curve c, where c is the top…

A: Given a thin metal wire in the shape of a semicircle occupies the curve C, where C is the top half…

Q: Consider a right circular cylinder open top tin can with volume 1000 cm. Suppose R and H are the…

A: Consider a right circular cylinder open top tin can with volume 1000 cm³. Suppose R and Hare the…

Q: 3. Give formulas for the volume and the surface area of acylinder with radius y and height s.

A: Formulas for the volume and the surface area of a cylinder with radius y and height s.

Q: Calculate the total mass of a metal tube in the helical shape r(t) = (cos(t), sin(t), 1²) (distance…

A: rt=&lt;cost, sint, t2&gt; (distance in centimeters)0≤t≤2πδx, y, z=9z g/cm

Q: Find the volume of the solid generated by re-volving the region in the first quadrant bounded by the…

A: First calculate the area ‘A’ under the given curve as.

Q: Find the area under one arch of the cycloid

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Q: 2. Let V be the volume of a right circular cone of height 10 whose base is a circle of radius 4…

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Q: Find the center of mass of a lamina as pictured below, with inner radius 3 and outer radius 5, with…

A: Given problem:-

Q: A lamina occupies the part of the rectangle 0 < x < 1,0 <y < 7 and the density at each point is…

A: Given: 0⩽x⩽1,  0⩽y⩽7ρ(x,y)=7x+8y+2 (A) Total mass is given by :M=∫01∫07(7x+8y+2)dydx

Q: Let V be the volume of a pyramid of height 20 whose base is a square of side 6. 20 6 Use similar…

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Q: The length x, width y and height z of a rectangular box are changing with time as x = 2+ sin 2t, y =…

A: length, x = 2 + sin2t width, y = 3 + cost height, z = 4 - sin3t   dVdt=? when t =π

Q: I f a right circular cone has a circular base with a diameter of length 8cm and volume of 16πcm³.…

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Q: The volume of the the sphere in the * figure is 5 نقاط = Va? - xZ a a dx

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Q: Find volume by evaluating vating triple integralspherical cordinates) use dp dodo Sin o

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Q: The lamina area density p has the shape the equations, find mass (m). y = vx , y= 0, x = 3

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Q: The volume of the solid given by tl

A: Given,             z=8-3x2+y2            z=x2+y22-2

Q: Find the center of mass of an arc y = f(x), where 3 -x + ,0<r<1 f(r) = In x 1<r< 2 2

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Q: O Use differentials to find a formula for the approximate volume of a thin cylindrical shell with…

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Q: Given a lamina that occupies the region outside the circle r = 2 and inside the 0, and has density p…

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Q: Consider a right circular cylinder open top tin can with volume 1000 cm3. Suppose R and H are the…

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Q: evaluate using substitution and then Integration by Parts.

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Q: Given a lamina that occupies the region outside the circle r = 2 and inside the cardioid r = 2 +…

A: Given a lamina that occupies the region outside the circle r = 2 and inside the cardioid…

Q: A lamina occupies the region inside the circle x + y but outside the circle x? + y = 1. The density…

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Q: Find the mass of the 4th quadrant portion of the disk x + y´ < 36 where the mass is given by p(r, 0)…

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Q: A trapezoidal plate with height h and bases a and 2a is submerged vertically in water as shown.…

A: We need to find integral.

Q: Verify the given moment(s) of inertia and find x and y. Assume that each lamina has a density of p =…

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Q: The linear density of a rod at a point x centimeters from the left end is, where the left 2 grams…

A: Given the density at a point 'x' cm from the left end ( origin) is d(x) = 2/(1+x) The length of the…

Q: Find volume if area enclosed is rotated about y= -2.

A: To find the volume if the area enclosed by the below curves is rotated about y = -2.

Q: B) By integration, determine the moment of inertia about the x-axis for the parabolic area shown in…

A: .

Q: Sketch the lamina S of constant density ρ = 3 g/cm2 occupying the region beneath the graph of y = x2…

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Q: y = cosx 10. Flnd the volume of a rotor that occurs when a curve is و و و و و و و و و و و می rotated…

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Q: 10 inches 6 inches 18 inches Using the dimensions from the previous item, find the rate of change in…

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Q: Find the Mass, Moments along x and y, Center of Mass, and Moments of Inertia of the region bounded…

A: The given region is bounded by x+y=2, x=0, y=0 as shown below: The region is given by R=x, y:…

Q: Use the shell method to calculate this volume in a different way. You should get that the volume…

A: Draw the curve y=e-x2, to identify the limits of integration from the region. Rotate the curve…

Q: Evaluate | dx approximately, using (a) the trapezodial rule, and (b) Simpson's rule, where the…

A: We need to evaluate ∫01dx1+x2 , here a=0 , b=1 , f(x) = 11+x2 We need to divide the interval in 4…

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.