In what option is the definite integral that calculates the volume of the rotating body formed by the function y = e ^ 4x, the line y = 4 and the region bounded by the oy-axis, rotating around the line x = ln2 / 2?

Q: Consider the region formed by the graphs of f (x)=sqrt( x) , f (x)= 1/x and x = 2. Set up the…

A: We sketch the region using a graphing calculator.  Red curve is : y=1/x Violet curve is: y=sqrt(x)…

Q: The region is bounded by x+6=0 and y^2+x+2=0 is revolved about x=-6. find the integral in finding…

A: Consider the provided question, The volume for solid of revolution between the curves x+6=0 and…

Q: )Find volume of body from above Z = 1- y² and below 2 = x² using Triple integration.

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Q: By using double integrals find the area between x = – y², lines y = x – 4, y = -2 and y = 2.

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Q: Write a definite integral for the area of the region bounded by y = 2 - x² and y = x2. Do not…

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Q: Write 2 integrals for volume of revolution about x axis  using washer method for region bounded by…

A: By using volume of a solid formula, we calculate the required volume.

Q: Find the center of mass of a thin plate of covering the region between the x-axis and the curve y=…

A: We have to find center of mass.

Q: Consider the region bounded by the graph of y = x^2 and the line y = 4. Express the definite…

A: a)

Q: The volume generated by revolving the area bounded by the curve y = x² and y = 4 around the line y =…

A: We can use the formula π∫abR(x)2dx to evaluate the volume.

Q: 5. Set up the integral to find the volume of the solid whose horizontal base is a circle of radius 2…

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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: 1. Find the volume generated by revolving the area bounded by the curve y = (1 – x²)² and the x–…

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Q: Set up the integral that gives the volume of the solid formed by revolving the region bounded in the…

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Q: Find the area bounded by the curve y = 1+x² and the lines y = 0, x = y and x = 2. Show complete…

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Q: Find the volume generated if the area bounded by the curve x² = y – 2 is revolved about the x-axis.

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Q: If we want to find the area between the curves x = −y 2 + 1 and x = y 2 − 1, shall we integrate with…

A: Given curves are x=-y2+1 and x=y2-1

Q: Only one of these calculations is correct to evaluate the volume of the solid generated by revolving…

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Q: the region bouded by x=y+2 and x=y^2 revolved about x=-1. Find the integral in finding the volume of…

A: The volume of the solid generated by revolving about the y−axis a region between the curves x = f(y)…

Q: Set up an integral to represent the total mass of a 2×5 rectangular sheet, whose density per unit…

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Q: Find the volume of the area bounded by the line y = 3x + 4 and the curve y = x² if it is revolved…

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Q: Evaluate the definite integral that represents the volume of the solid formed by revolving the…

A: Explanation of the answer is as follows

Q: 5. Find the volume of the solid generated by revolving the area bounded by the curve y = 4x - x2 and…

A: We are asked to find volume of the solid

Q: 2 Set up integral for the volume about y-d y=- x u X- y-2.

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Q: Sketch the region of integration for the double integral given by integral 0 to ln2,e^x to 2 dydx…

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Q: Find the volume formed by revolving about the x-axis the area bounded by the curve (x + 1)^2 = 4(x -…

A: Given data:- Equation of curve, x+12=4x-1 Line, y=4

Q: Find the volume generated by revolving about the line x=0 the area bounded by the curve y^(1/2) =…

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Q: The area bounded by the curve y^2 = 12x and the line x = 3 is revolved about the line x =3. What is…

A: Recall: Formula for volume,    V=π∫ab(radius)2dy Given,  y2=12x Substitute x=3 in the above…

Q: Sx√x +Tdx

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Q: Find the mass M and center of mass x of the linear wire covering the given interval and having the…

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Q: If a double integral yields a negative value, does this mean that the volume is also negative?…

A: Topic:- double integration

Q: Set up an integral for the volume obtained when the region is rotated about the line x = -3. DO NOT…

A: To set up the integral to calculate the volume of the solid obtained by rotating the below region…

Q: If the region is enclosed by y=x^2 and y=2x, find the exact area by a definite integral

A: Answering the question for the area enclosed between 2 curves as the 1st question is incomplete.…

Q: 5. The area bounded by the curve y = 3x – x3 and the line y = 2 is revolved about the line y = 2.…

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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: The volume of the solid rotating a curve about y-axis is given by the formula, V=∫cd πx2 dy  where c…

Q: Find the volume generated by revolving about the line x=0 the area under the arch of the sine curvey…

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Q: THE REGION BOUNDED BY y = 205x - 1 AND THE X-AXIS IN THE INTERVAL πT 317 EX2 IS REVOLVED ABOUT 2 2…

A: The Shell formula for Revolution about a vertical line is used to evaluate the volume of the solid…

Q: The region bounded by the curves y = x² – 2x and y = 4x is revolved about the line y = -1. Which…

A: The given curve is y=x2-2x and y=4x. Now, find the intersection point of these two curves.…

Q: Evaluate the double integral that will solve for the area of the region bounded by y = cos(x) + 5, y…

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Q: Determine the volume generated when the area bounded by the curves y-sinx and y%3cosx from x-0 to…

A: Aim: We need to find the volume of the solid obtained by revolving the bounded region (light yellow)…

Q: Evaluate the double integral that will solve for the area of the region bounded 2. by y = cos (x) +…

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Q: Find an approximation of the area y = 16 - x² over the interval [0,4] using L6 (left end point…

A: Use: Area=∆xf(x1)+f(x2)+f(x3)+f(x4)+f(x5)+f(x6)       where ∆x=b-anwhere n= no. of rectangle…

Q: The area bound by the curve y^2 = 12x and the line x = 3 is revolved about the line x = 3. What is…

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Q: A solid is generated by revolving about the x-axis the region in the first quadrant bounded by the…

A: It is given that, y=1+sin2x,x=π2   Volume of solid formed by revolving the first quadrant region…

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: Sketch and set up integrals for volume of the region bounded by y = x^2 + x - 2 and y = 3x + 13…

A: Please see the below pictures for detailed solution.

Q: Find the volume generated by revolving about the line x - 4 = 0 the area bounded by the curve x2 =…

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Q: Solids of Revolution ( refer to the image )

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Q: The volume of the solid obtained by rotating the region enclosed by y = x, x = y about the linex =…

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Q: The volume of the solid obtained by revolving the region bounded by the curve y = x - x² and the…

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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.