The region between y = x and y = 3x is rotated around the line x = 4. %3D Using the shell method we find the volume of the resulting solid by solving the integral: dx Solve the integral to give the exact value of the volume: Volume =

Q: The form SoE – 2) sin zdæ + S – x) cos ædx - represents the volume of the solid S generated by…

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Q: Given f[(x - +sin y) dA, where R is the region bounded by x= 0, x=2, y=0, and R y=x. Sketch the…

A: !st we will draw graph of given equation and identify region of integral  For type1 region  y will…

Q: e 2 ху dy dx 1 In x

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Q: y= md y=x+2 ... and Revolve the area about the x-axis. Using circular ring method, the volume of the…

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Q: Given [[(3xy)dA, where R is the region bounded by y=0 and y = /9-x² . Sketch R the region of…

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Q: 10. Write the integral needed to find the volume of the solid formed when the graph of the region…

A: Refer to the figure shown below

Q: Each integral represents the volume of a solid. Describe the solid. a) S 2nx cos x dx COS X t/2 b)…

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Q: 14. If a is a positive constant, use a triple integral to determine the volume Q between z = a and…

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Q: b) consider the area between the X-aXis and the curve y= Sin2x for Os X<#/2. when this area is…

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Q: Each integral represents the volume of a solid. Describe each solid

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Q: 1. Write a definite integral to calculate the area of R. 2. If the line y = c divides R into two…

A: For the given curves, the definite integrals will be: PART 1 For finding area , A=∫x=03f(x)-g(x)…

Q: Using a single integral, which method/s can be applied to determine the volume of solid generated by…

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Q: (6) Find the volume of the solid obtained by revolving the region between y = y = e¬¤ for 0 <x < ÷…

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Q: A solid S is generated by revolving the region enclosed by the line y = 2x + 1 and the curve y = x²…

A: Given:  y=2x+1  ...(1) ;     y=x2+1    ...(2)(a) To find the limits for x-values:Put (2) in (1) ⇒…

Q: Evaluate the iterated integral. (x² + y² + z² ) dx dy dz Exact number, no tolerance. 8 9 (x2 + y? +…

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Q: Given the integral dx Use the integral table to help you find the Anti Derivativ

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Q: Which integral below gives the volume of the solid of revolution obtained by rotating the bounded…

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Q: valuate the integral by reversing the order of integration. r/2 cos(x) V (16 + cos?(x) dx dy…

A: In this question we will use basic concepts of calculus i e concepts of double integration. We will…

Q: Hence, find the area of the specified region using integration. The base of a solid is the region…

A: According to the given information, it is required to find the area of the specified region using…

Q: SET UP the integral necessary to find the volume of the solid for the shaded area revolved about the…

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Q: Given || (x² + sin y)dA , where R is the region bounded by x=0, x=2, y=0, and y =x. Sketch the…

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Q: Each integral represents the volume of a solid.Describe the solid.

A: Given ∫0π22π cos2(x)dx

Q: For the integral 2xy dy dx, -2 J-V16-x² sketch the region of integration and evaluate the integral.…

A: Given : Integral ∫-20∫-16-x20 2xy dy dx   To find : Sketch the graph of the region of integration…

Q: Consider the region T bounded by the graphs of y = x2, y = -3x, and x = 4. (a) Find the area of T. X…

A: Given that the region T is bounded by the graphs of y=x2, y=−3x and x=4. a We have to find the area…

Q: 1. For the integral /| av dy dr dz , do the following. AP dV = E a) Evaluate the integral. b) Sketch…

A: (a) (C)

Q: 2. Given (x - y) dy e dr. Reverse its order of integration and evaluate. Also, draw a sketch of the…

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Q: A solid is formed by rotating the area between the curves y = x and y = x² between x = 0 and x = 1…

A: To solve this problem we use washer method.

Q: The region in the first quadrant bounded by the graph of y=8-x3/2, the x-axis, and the y-axis is…

A: We need to choose correct option.

Q: (d) m | (2)² – (sin(x))² d.x •T/2 (e) 2nx cos(x) dx 2 (f) T((4y)? – (3³)?) dy

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Q: The region bounded by y = cos x - 1 and the x-axis in the interval 3πt ≤x≤³ is revolved about the…

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Q: y=x nd y=x+2 Revolve the area about the x-axis. Using circular ring method, the volume of the solid…

A: The given curves are; y=x2 and y=x+2. And the bounded region is as shown below; Apply the circular…

Q: Now we can say that the volume of the solid created by rotating the region under y = 4e- and above…

A: Given that y=4e-x2, y=0, x=0, x=1. At x=0, y=4 At x=1, y=1.472. Given that the region is rotated…

Q: 3. A solid is generated when the region in the first quadrant enclosed by the graph ofy=(r +1)', the…

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Q: Applying the formula of integration by parts, evaluate the following integral SE sin? 2 (Maksimum…

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Q: 2 |[ v'dydx, (a) Evaluate the given integrals (b) Interchange the order of integration and evaluate

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Q: The graph above shows how the x-axis, the y-axis, the line y = -x + 3 and the curve %3D y = 2 sin x…

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Q: The solid E bounded by the equation 9x^2 + 4y^2 + z^2 = 1 and located in the first octant is…

A: Given : Equation of sphere is 9x² + 4y² + z² = 1

Q: The region between the graph f(x) = Vĩ and the x-axis for x E [0, 4] is revolved about the line y =…

A: We know that volume by washer method

Q: Each integral represents the volume of a solid.Describe the solid.

A: Given π∫25y dy=πy2225=π252-42=212π This is the area between two circles that have radii 5 and 2.

Q: Evaluate dy dx by changing the order of x Vx4 + y? integration. Sketch the region of integration.

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Q: (1 – x*)² – (1 – Væ)°] dx a) Describe a solid of revolution whose volume is equal to the given…

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Q: Given a region R enclosed by the curves: y = 1 + sin(™), y = 4(x and x = 2 as shown above. (x-2)² 4…

A: The region R is enclosed by the following curves: y=1+sinπx4y=4-x-224x=2 We are asked to find the…

Q: For the given double integral, sketch the region of integration and write an equivalent double…

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Q: F-(2x-3)も- 2xyj-42kノexauate SSS v.P dv, x 0,yo, 2, 2×4 2ytz=4

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Q: The volume of the solid obtained by rotating the region enclosed by 4x y = e* + 2, y = 0, 0, X = 0.1…

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Q: Given f[(x* + + sin y) dA , where R is the region bounded by x=0, x=2, y=0, and y=x. Sketch the…

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Q: (10,00 Puanlar) 3- By using the cylindrical shell method, determine the integral which gives the…

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Q: A) Sketch the region over which integration takes place, then reversed the order and Evaluate one of…

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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.