18. The integral that represents the volume of the solid obtained by rotating the region bounded by the curve y = 7e-3x and the lines y = 0,x = 6 about the x-axis is: 98n e-6x2 dx a b 987 -3x* dx 49n e-3x+ dx d 49n -6x² e 14п -3x² dx e

Q: Consider the following. y = 6 cos¹(x), y = −6 cos^(x), - 7 = ≤x≤ about x = π 2 (a) Set up an…

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Q: B is the region bounded by the following graphs: 1 A1: ]y – 1| = ;x A2: (y – 4)² = –4(x – 6) %3D…

A: According to the question,Given curve,A1 : y-1 = 12x,A2 : y-42 = -4x-6

Q: Find the volume of the solid generated by revolving about the line y =-1 the region above the X-axis…

A: Here, for x = -1 to x = 0,  Area is bounded by  the curves 2y = x + 3  and  y2 = - x

Q: (2 points) The volume of the solid obtained by rotating the region enclosed by y = x², y= 5x, about…

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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: Consider the region bounded by the graphs of y=In x, x-1, and y=1. Which of the following integrals…

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Q: 2. Consider the region bounded by the graphs of y = 3* – 1 and x = y² – 3, illustrated at the right.…

A: The objective is to setup the integral that calculate the center of mass of the region bounded by…

Q: 2 Consider the region berween the grsph of fo and the line y-2on the interval (0. IL Use diwaher…

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Q: 8 Use the method of cylinders to determine the volume of the solid obtained by rotating the region…

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Q: Which of the following integrals represents the volume of the solid obtained by rotating the region…

A: In the question we have to find the integral representation of the volume.

Q: 3) Calculate the volume of a solid obtained by rotating the region bounded by: (Any method) y = x-x²…

A: To calculate the volume of the solid obtained by rotating the region bounded by below equations…

Q: The volume of the solid formed by revolving the region bounded by the graphs of y = vx + 3, y = 0…

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Q: Find the volume of the solid obtained by rotating the region bounded by x = -6+ y² and a y about the…

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Q: The area bounded by y =3Vx and 2y =3x is revolved about the x-axis. Using cylindrical shell method,…

A: Solve the following

Q: 5. Let S be the solid in the first octant bounded by the cylinders x² + y² = 4 and x² + z² = 4. (a)…

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Q: 5- Find the volume of the solid generated when the region bounded by the curve x cos y and the lines…

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Q: 2.4\(a)Use a double integral to determine the volume of the region formed by the intersection of the…

A: Volume between cylinders using double Integrals

Q: The integral: 2nx coS x dx represents the volume of a solid obtained by rotating the region: A)…

A: We have to find the region that is rotated.

Q: Consider the region bounded by the curve y = x2and y = 4x. a) Set up (but do not evaluate) an…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Consider the region on the 1st quadrant bounded by y = √4x²,x and y- axes. If the region is revolved…

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Q: 4.Find the volume of the solid that is obtained by rotating the region bounded by y = /x +1, y = Vx…

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Q: SI xyz dW, B is region bounded by two cylinders, wwwm w В x^2 + y^2 = 16, x^2 + z^2 = 16, and the…

A: Consider the given volume of the two intersecting cylinders given by  x2+y2=16 andx2+z2=16  as shown…

Q: Set up an integral for the volume of the solid generated by revolving the region in the first…

A: We can see the region bound by the curves. Now we will set up the integral.

Q: 4. Find the volume of the solid obtained by rotating the region bounded by the given curves about…

A: Given that y=ln x, y=1, y=2, x=0 find the volume about y-axis V=π∫12x2 dy

Q: y = x,    y = 4x − x2,    about the line x = 8 Set up an integral for the volume of the solid…

A: Draw the diagram as shown below figure and find out the intersection points.

Q: 5. Find the volume of the solid form by the region bounded by the curves y = 12 – 4x, y = 8x 12, is…

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Q: 3. Determine the volume of the solid obtained by rotating the region bounded by y = 2vx – 1 and y =…

A: Note: according to our guidelines we can answer only third question because student asked that…

Q: Set up the integral of the volume of the solid formed by revolving the region bounded by y = x² and…

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Q: The area bounded by y=3Vx and 2y =3x is revolved about the y-axis. Using circular ring method, set…

A: This question can be solved using the concept of circular ring method.

Q: The volume of the solid obtained by rotating the region enclosed by 1 x² y = with limits of…

A: We have to find the volume of the solid.

Q: 3. Consider the region bounded by y = 5 - x 2, y = x - 1, and x = 1. Set up an integral to find the…

A: Concept Used: (Shell method) The Volume of the solid obtain by the rotating the region is given by…

Q: If the region under the graph of +++/-/1 -- y = x² + from x = 1 to x = 5 is rotated about the…

A: NOTE: Refresh your page if you can't see any equations. . the given function is  and the given…

Q: 10- By using the disk method, determine the integral which gives the volume of the solid obtained by…

A: Consider the equation of the curves, y=e-x, x=2 and x-axis, and y-axis. The graph is shown below,

Q: Consider the region bounded by the graphs of x = 1/2, x = pi, y = ln 2x, y = ex,. Set up a definite…

A: Given: Considering the region bounded by the graph of . To find the volume of the solid that is…

Q: Compute the volume of the solid of revolution obtained by rotating the region bounded by y| = 5x, x…

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Q: 6. Find the volume of the solid obtained by rotating the region bounded by the given curves about…

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Q: 12. Set up an integral to find the volume of the solid generated when the vegion bounded by y=…

A: Giveny=x  , y=o , x=4

Q: The volume of the solid obtained by revolving the region enclosed by the curves y = 7 - x2, y = 0, x…

A: Given: The region is bounded by following curves: y=7-x2x=0y=0x=2 And an axis of revolution is x=3.

Q: Set up the double integral that finds the volume of the solid under the surface z = 2x2 + 2y over…

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Q: The volume of the solid obtained by rotating the region bounded by y = a?, 6x, about the line x = 6…

A: We rotate the shaded region around x=6 lineExpress the given functions as function of y.

Q: 12. Use the method of cylindrical shell to determine the volume of the solid obtained by rotating…

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Q: The volume of the solid obtained by rotating the region enclosed by 1 y = 0, x4 y = x = 3, and x =…

A: Plot the given curves y=1x4, y=0, x=3, x=6 on a co-ordinate plane and obtain the region enclosed by…

Q: The solid obtained by rotating the region bounded by the curves y = 2x and y = 3x between x =D0 and…

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Q: 6. Find the volume of the solid formed by revolving the region bounded by the graphs of y = x2, x =…

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Q: 2. Consider the region bounded by y 5- x 2 , y = x- 1, and x = 1. Set up an integral to find the…

A: Region is bounded by y=5-x2y=x-1x=1 region is revolved about x axis

Q: 5. Let S be the solid in the first octant bounded by the cylinders x² + y² = 4 and x² + z² = 4 (a)…

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Q: Find the volume of the solid obtained by rotating the region bounded by x = -8+ y² and x = 2y about…

A: Given : x=-8+y2 , x=2y  and x=-9. Therefore, -8+y2=2y⇒y2-2y-8=0⇒y2-4y+2y-8=0⇒yy-4 +2y-4 = 0⇒y-4y+2 =…

Q: 2. Find the volume of the solid generated by rotating the region bounded by y =÷x³ and %3D y = 6x -…

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Q: Consider the region in the first quadrant bounded by y = x', the y-axis, and the line y = 1. Set up…

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Q: The volume of the solid obtained by rotating the region bounded by y = 2, y = 2x, about the line = 2…

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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.