The given curve is rotated about the y-axis. Find thearea of the resulting surface.   x =  1/4 x2 - 1/2 In x    1≤x≤2

Q: In  the given equation as  follows , find the area of the surface generated by revolving the curve…

A: Given: x=t ; y=4-2t; 0≤t≤2To find the area of the surface generated by revolving the curve about…

Q: Consider the region enclosed by the curves y = r² + 1 and y = 2r + 1:

A: Consider the region bounded by the curves y=x2+1 and y=2x+1 The intersection between the curve is,…

Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y =x2 4 In(x),…

A: Given curve is y=14x2-12ln(x) Differentiate with respect to x. dydx=x2-12x

Q: Find the area of the surface generated when the given curve is revolved about the given axis. y =…

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = Vx, 1sy& 4

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Q: Find the area of the surface generated when the given curve is revolved about the given axis. y = 3x…

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Q: | (4x – y? + 6z)ds

A: We have to find the line integral of the function : 4x - y2 + 6x Along the curve C , which is the…

Q: Find the area of the surface generated when the given curve is revolved about the given axis. y= ,…

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Q: Find the exact length of the curve. = 글Vy (y-3), 16 sys 25 X =

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = Vx, 3sy54

A: Given:y=x3  ,     3≤y≤4Surface area :S=∫ba2πx 1+dxdy2   dy

Q: Consider the curve C shown in the attached figure, which is the intersection between surfaces S, and…

A: To parameterize the curve which is the intersection of the below curves in the first octant.…

Q: 4. Graph the surface 1 = y² + z² – by first sketching at least three cross sections in each of the…

A: Graph the surface by using a 3D graphing calculator like desmos. Three cross sections in xy plane…

Q: 4. Let C be the unit circle in the xy-plane, and S be the boundary of the square with side 1,…

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Q: Find the area of the surface generated when the given curve is revolved about the y-axis. y-(3x)13,…

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Q: 3) Find the area of the surface obtained by rotating the given curve about the indicated line, y=x2,…

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Q: Find the area of the surface generated when the given curve is revolved about the given axis. y = 12…

A: This question is based on application of Integration.

Q: for y^2 - z^2 =1, find the point(s) on surface nearest to origin

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Q: Given : The region is bounded by the curve y=4x-x^2, y=4 and the y-axis Problem : The intersection…

A: We have to determine the points of intersection of the given curves and find the value of a+b+c.

Q: (a) Find the area of the surface generated by the given curve about x-axis y = Vx , 1< x < 4

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Q: Given the expression x⁴-3x+7=y³ on the interval [3,5], determine the surface of revolution if it is…

A: Given that, the expression is x4-3x+7=y3.

Q: Find the area of the surface generated when the given curve is revolved about the x-axis.  y =(5x -…

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Q: Find the area of the surface generated when the given curve is revolved about the x-axis. X₁ 1 y=6 1…

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = x2 - 1 -…

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Q: Find the area of the surface generated when the given curve is revolved about the x-axis. y = V3x +5…

A: Surface area of the bounded region along x axis

Q: Consider curve C which is the intersection of the surfaces shown in the attached figure. image 1…

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Q: Find the curvature K of the curve at the point P. r(t) = 4ti + 5t²j, P(−4, 5)

A: Given curve r(t)=4ti+5t2j We need to find the curvature at point P.

Q: Find the area of the surface generated when the given curve is revolved about the x-axis. y = V3x +…

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Q: 4. Find the area of the surface generated when the given curve is revolved about the y-axis. y=(2x)"…

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: Consider curve C which is the intersection of the surfaces shown in the attached figure. x = y² +3…

A: Given is the curve. The object is to approximate the given integral.

Q: /| 9(x + y) dA where R is the region that lies to the left of the y-axis between the circles x2 + y?…

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Q: 1. Find the exact length of the curve. (a) y² = 4(x+ 4)°, 00 (b) y = In z, 1<I<2

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Q: The length of the curve y = Ey3/2 between x = 0 and X 1 is %3D

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Q: The graph of y =√x from (1, 1) to (4, 2) is revolved around the x-axis. Find the area of the…

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = 0SxS 12

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Q: 5. Find the exact area of the surface generated by revolving the given curve about the y- axis, Draw…

A: Application of integration

Q: Find the area of the surface generated when y² = 12x is revolved about the x- axis from x = 0 to x =…

A: Given curve,y2 = 12xRevolved about x-axis from x=0 to x=3.

Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. 1 In(x), 4< x <…

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Q: Find the length of the curve y = a +; from a = 0 to x = 1. O None 5/2 O6/2 3 O4/2 3. 3

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Q: ind the exact length of the curve. X = 8 4y2' 1sys 2

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. = Vx, 1sys 5

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Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y = =x² - 1 1,2…

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Q: Consider the intersection curve C in the first octant, between the surfaces s1: 2 = 1- r? S2 : 2 +…

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Q: Find the area of the surface generated when the given curve is revolved about the given axis. y = 12…

A: Consider the curve y=12-3x, 1⩽x⩽3. Surface area of solid when curve is rotated around x-axis is…

Q: Find the area of the surface generated when the given curve is revolved about the x-axis.  y =…

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Q: Consider the region in the fourth quadrant between the circles x² + y² = 4 and x² + y² = 25 and…

A: answer is in next step. please give a like !!

Q: Evaluate | 9+x* dx+xy*dy along the positively oriented curve C where the C is the boundary of the…

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Q: 3. y = Vx,0< x<1 ; у-ахis

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Q: Y=Ğ jfor exS3; about The X-axi's

A: The formula for the surface area generated by rotating the curve  A=2π∫abf(x)a+(f'(x))2dxHere…

Q: Find the area of the surface generated when the given curve is rotated about the x-axis. y = 8/x on…

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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.