Suppose that f is a continuous function on [a, b], and let R be the region between the curve y = f(x)and the line y = k from x = a to x = b. Using the method of disks, derive with explanation a formula for the volume of a solid generated by revolving R about the line y = k. State and explain additional assumptions, if any, that you need about f for your formula.

Q: sketch the solid obtained by rotating the region underneath the graph of f over theinterval about…

A: We sketch the region first The shaded region is rotating about the line.  After rotation, the solid…

Q: Suppose f is continuous on [a, b] and the graph of f lies above the line y = c. Let V denote the…

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Q: Let f(x)=√√3x-2 on the interval volume of the resulting solid. 83л -unit3 135 43x 125 83x 125 43x…

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Q: A region in the xy-plane has a z-output given by the equation z = In(x") + 2xy - e*y. Detemine the…

A: The double integral of the function  will result the volume of the solid.

Q: Consider the region R bounded by the graphs of y = ƒ(x) > 0, x = a > 0, and x = b > a. If…

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Q: Compute the volume of the solid w w w w ww obtained by revolving the graph of the function f(x)=7x°…

A: Volume of solid obtained by revolving is given as Volume=∫ab πfx2dx

Q: Let f(x) = log(x + 1), g(x) = 10-x, and R be the region bounded by the graphs of f and g, as shown…

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

Q: Find the volume of the solid generated when the region between the graphs of the equations f(x) + x…

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Q: The volume, when the curve y=v(3+ x^2 ) from x = 0 to x = 4 is rotated about x - axis is 100n 100/9…

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Q: Define R as the region bounded above by the graph of f(x) = x and below by the graph of g(x) = x²…

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Q: The volume generated when the area of the finite region enclosed by the x-axis and the curve y = x –…

A: Given curve is y = x-x^2

Q: If R denotes a region bounded above by the graph of a continuous function f(x), below by the x-axis,…

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Q: Find the volume of the solid generated by the curve f(x)=4-cosx revolved 360° about the x- axis in…

A: We calculate the required volume using integration. ∫4-cos x2dx=∫16-8 cos x + cos2 xdx=∫16-8 cos x…

Q: Let f(x) = 25 - x2. A solid is generated by rotating the reagion bounded by f(x), the x-axis and the…

A: We first sketch the region using a graphing calculator  the shaded region is rotating about the y…

Q: Suppose that a thin metal wire in the shape of a semicircle occupies the curve c, where c is the top…

A: Given a thin metal wire in the shape of a semicircle occupies the curve C, where C is the top half…

Q: 29. Suppose f(x, y) is continuous over a region R in the plane and that the area A(R) of the region…

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Q: A solid of revolution is formed by rotating a curve y = (x) about x-axis between the planes x = 0…

A: Given values are:

Q: Let f be continuous and let Bg be the ball of radius ɛ centered at the point (xo, yo, zo). Let vol…

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Q: Consider the functions f(x) = x° + 2 and g(x) = x² + 2x + 2. Let R be the region between the graphs…

A: Given that, the functions are fx=x3+2 and gx=x2+2x+2. Note that, The volume of the solid formed by…

Q: Suppose f is continuous on [a, b] and the graph of f lies above the line y = c. Let V denote the…

A: Rotating the region between two curves y=f and y=c between x=a and x=b around a line of the form y=c…

Q: Given that R is the region bounded by the curve y = x+3 and line y = x-3. (c) By using the limits in…

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Q: Suppose that R is the finite region bounded by f(x)=√x and f(x)=x3. Find the exact value of the…

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Q: Define R as the region bounded by the functions f(x): = 3 and g(x) = x + 1 between x = 0 and x = 1.…

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Q: Suppose R is the shaded region in the figure, and f(æ, y) is a continuous unction on R. Find the…

A: The given double integral is ∫AB∫CDfx,ydy dx. The region of integration is R:A≤x≤B,C≤y≤D We need to…

Q: Given a curve having an equation of y2 = 4x. a) Compute the area bounded by the curve and the line…

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Q: Define R as the region bounded by the graphs of f(x) = 6x - 5x² and the x-axis. Which of the…

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Q: Compute the volume of the solid the lies below the surface z = f (x, y) = x+2y? and above the…

A: Given surface z=fx,y=x+2y2. Plotting rectangle R=[0, 2]×[0, 4]. Dividing rectangle into 4 equal…

Q: Let f (x) = 9x and g(x) = x2 + 6x, and let R be the bounded region between the graphs of f and g.…

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Q: What is the exact volume of a solid that is formed by rotating the area between the functions f (x)…

A: Given: To find the volume of the region between functions f(x)=5x and g(x)=3x rotating about the…

Q: Suppose that R is the finite region bounded by y = x, y = x + 1, x = 0, and x = 2. Find the exact…

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Q: The surface of an island is defined by the following function over the region on which the function…

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Q: Suppose that R is the finite region bounded by y = x, y = x + 2, a = 0, and a 3. Find the exact…

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Q: Given a solid bounded by f(x,y) = 3x² + 2y“, xy-plane, x = 3, x = 12, y = 6 and y = 16. If the order…

A: find the volume using double integration.

Q: Given a solid bounded by f (x, y) = 5x2 + 4y", xy-plane, x = 4, x = 9, y= 5 and y = 15. If the order…

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Q: and C be the boundary curve of the region R= {(x,y)ER²:x²+ s1.y20) oriented counter-clockwise, where…

A: Here, the field is F(x,y)=x2y2-2βπy, x2y2+x2 and boundary curve C is as shown below: The curve C is…

Q: Let f (x) = cos x and g(x) = sin x. Find the center of gravity (x, y) of the region between the…

A: We are given the function,f(x) =cos x and g(x)=sin x. And we need to find the center of gravity (x,…

Q: Find the exact volume of the solid generated when each curve is rotated through 360° about t x-axis…

A: Explanation and solutions is given below....

Q: Let f(x) = 25 - x2. A solid is generated by rotating the reagion bounded by f(x), the x-axis and the…

A: Find the f(y) and g(y).

Q: Assume ƒ and g are continuous, with ƒ1x2U g1x2 U 0 on 3a, b4. The region bounded by the graphs of ƒ…

A: fx≥gx on a, b. The region bounded by these functions and the lines x=a & x=b is revolved about…

Q: Consider a thin rod oriented on the x- axis over the interval [0, x/2]. If the density of the rod is…

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Q: a) Show that the volume enclosed when revolving the curve y = f(x) - where f: [a, b] → [0, 0) -…

A: Note- In the given question, it has to be V = π∫abf2x dx instead of V = π∫abf2x x. So, we are…

Q: The volume of the solid that results when the region between the curve y=ex and the line y = 0, from…

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Q: What is the exact volume of a solid that is formed by rotating the region, enclosed by the functions…

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Q: Define R as the region bounded by the functions f(x) = 2 and g(x) = e¯i between x = 0 and æ = 2. If…

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Q: What is the exact volume of a solid that is formed with a square base, aligned with the center of…

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Q: Define R as the region bounded by the graphs of f(x)=x2 +4x+5, x=−3, x=0, and the x-axis. Using the…

A: Please see the diagram on the white board. The line in blue is the plot of the given curve f(x)=x2…

Q: Let f be continuous and nonnegative on [a, b], and let R be the region that is enclosed by y =…

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Q: The volume, when the curve y = √3+ x2 from x = 0 to x = 4 is rotated about x-axis is O100m cu units…

A: Given:    y=3+x2 from x=0 to x=4  Volume of the curve V=π∫abfx2dx

Q: Suppose f is continuous on [a, b] and f(x) 20 for all x € [a, b]. The integral T| [F(x)]²dx is 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.