Consider the region bounded by the graph of y2 = x and the vertical line x= 4. Express the integral representing volume of the solid obtained by rotating the bounded region (b) (i) about the y-axis using the washer method; (ii) about the line x=5 using the shell method. %3= Do not evaluate your definite integrals.

Q: Draw the area bounded by function y = x^2 + 1 and line y = 5, then calculate the volume of the solid…

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Q: Find the volume of the solid generated by revolving the region by graph of function f(x) 10 ;x 1.25,…

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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: Find the volume of the solid obtained by revolving the region bounded by the graph of f(x) = 8e-lx,…

A: Given the curves: fx=8e-7x, x axis, y axis, x =6rotated about y axis

Q: Find the volume of the solid generated byrevolving the region in the first quadrant bounded by the…

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Q: The area bounded by the curve y = x,y% = 2x and y = 4 is revolved about the line x =-2 Determine the…

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Q: In what option is the definite integral that calculates the volume of the rotating body formed by…

A: Given, y=e4x, y=4 , y-axis is region  it is rotating around the x=(ln2)/2 parallel to y-axis. so…

Q: Q4) Fid the volume of the solid generated by revolving the region between the curve y = x + 1 and…

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

A: a)

Q: What is the volume of the solid whose 1 30 cross-sections are squares perpendicular to the x-axis…

A: Given: The region bounded by the curves are: y=x and y=x The cross-sections are squares…

Q: π Find the area of the region enclosed by y = sin¯¹(x), y = and the y-axis without using integration…

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Q: Given the area in the first quadrant bounded by the curve x2 = 8y, the line x = 4 and the x – axis.…

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Q: Find the volume of the solid obtained by rotating the region bounded by x = -3 + y² and x = -2y…

A: Given the curve is x=-3+y2, x=-2y Outer radius=-3+y2+9=y2+6 inner radius=-2y+9 Intersection point is…

Q: The volume obtained by rotating the region bounded by y=36-(x-3)2 and the x-axis about the line x=…

A: Given Region bounded by curve  y=36-(x-3)2, x-axis  (means y=0) about line x=10. the maximum y is…

Q: Find the volume of the solid of revolution generated by revolving the region bounded by the graphs…

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

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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: The area defined by the curves: y= 10/ square root (x2+1)    ;     x=1, x=4, y=1 is rotated around…

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Q: Find the area of the region enclosed by the curves y = x² and y = x by integrating with respect to…

A: Let f(x)=y=x2      (1)g(x)=y=x          Compare x=x2x2-x=0x(x-1)=0x=0 or x-1=0x=0 or x=1Put value of…

Q: Find the volume of the solid generated by revolving about the x-axis, the area bounded by y = x3,…

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Q: y = 9x² and y = 0 about x = −4

A: Given:        y=9-x2 and y=0         x2=9           x=3 about x=-4

Q: To find the volume of the solid generated by revolving the area bounded by the given curve about the…

A: The volume of the solid generated by revolving the area bounded by the curves f(x) and g(x) over the…

Q: Find the volume of the solid generated from revovling the area of the region bounded by the curve y…

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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: Find the area of the region enclosed by the curves y = x² and y 4.x by integrating with respect to…

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Q: Find the volume of the solid whose base is in the first and second quadrant(s) bounded by y = 32 –…

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Q: Set up the integral for the surface area of the solid created by revolving y= Vx 1< x <8 about the…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Find the volume if an arch of y=sin⁡2x (bounded by the x-axis) is revolved about the x-axis

A: Volume of revolving  of y = f(x) about X-axis  Vol =∫abπ(f(x))2dx

Q: Find the surface area of revolution about the x-axis of  y= 4sin(4x), from x is greater than or…

A: Equation of curve is y= 4sin(4x) The interval is 0≤x≤π2

Q: Find the volume of: the area bounded by the curve y2 = 9x and the line x = 4 revolved about x = 4.

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Q: The area bounded by the parabola y² + x - 2y = 1, the x — axis, and the line x = 2 revolves about…

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Q: Find the volume of the solid generated when the area bounded by the curve y^2=x, x-axis and the line…

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Q: What is the volume generated when the area bounded by y=2x2+1; x=1; x=2 is revolved about the line…

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Q: From the equation of x² + y² + z² = 25 and inside the equation x² + y² = 9 Calculate the volume of…

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Q: Set up the equation in solving volume of the the area bounded by x = /y, x = 0, and 3 revolution at…

A: Given that: x=y, x=0, y=3    and revolution around y-axis.   NOTE: As per our answering guidelines,…

Q: The area bounded by the line x=1 and the curve y^2-x=3 is revolved about the line y=4. Using Shell…

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Q: Find the volume generated by the region bounded by the curve 4x = y² and the line x - 4 = 0, rotated…

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Q: 3.Find the volume of the solid formed by revolving the region bounded by the graph of f(x) = vsin x…

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Q: Find the volume of a solid obtained by revolving the curve x2=y-3 about x-axis from x=1 to x=3

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Q: Solve for the volume of the solid generated by revolving about the y-axis the area under y x = 4 to…

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Q: Find the volume of the solid generated by revolving the region enclosed by the graphs of y =…

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Q: Find the volume generated by rotating the region in the first quadrant bounded by y=e^x and the…

A: Answer

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

A: Given the region is bounded by the curves x=2y and x=8-y2 about the line x=9. The curves intersect…

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

A: Solution :-

Q: Find the volume of the solidgeneruted by revolving the region between the x-axis and the curve…

A: We first find where y=x2-2x and x-axis meets.x-axis means y=0.So, 0=x2-2xor, 0=x(x-2)or, x=0, 2 The…

Q: . The area bounded by the curve y = 3 – 2x + x² and the line y = 3 is revolved about the line x = 5.…

A: The volume of the solid of the revolution can be calculated using the shell method. The formula to…

Q: Find the volume of the solid of revolution generated by revolving the region bounded by the graphs…

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

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Q: Find the volume of the solid formed by invoving the region by the graphs y=x^3 , x = 1 , about x = 2

A: We first sketch the region using a graphing calculator.

Q: Q3. The region bounded by the curve y x² and the line y = 3x, is revolved about the line x = -3.…

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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.