Let R be the region in the first quadrant bounded by the curves. Sketch the region enclosed by the given curves. Then calculate the volume obtained by rotating the region about the x − axis, and about y − axis. a) y = x^3 and y = 2x − x^2 b) y = (x +1)^(1/2) and y = x^2

Q: 2. Express the radius of a sphere as a function of the sphere's sur- face area. Then express the…

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Q: 1. Determine the volume of a solid of revolution generated by rotating around the x-axis the region…

A: Since you have asked multiple questions here, we will solve the first question for you. To get the…

Q: Write the equation for the volume obtained by rotating the region shaded below about the line y =…

A: Hee, the volume formula is: V=π∫ab((f(x)-a)2-(g(x)-a)2) dx where the region bounded by the curves…

Q: Calculate the volume of the solid formed by revolving about the line y=1 the region bounded by the…

A: Integration is the process of finding the anti-derivative function of a given function. The…

Q: Find the volume of the solid generated by revolving the area bounded by x- y² + 4y-4 0 and x- y-4 0…

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Q: the area bounded by the parabola y ² =4ax, the y-axis, and the line y=2a revolves about the line…

A: given that the region is bounded byy2=4ax and y=2a and  revolves about y=2a Graphically,

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

A: Given a solid obtained by rotating the region bounded by the following curves:

Q: Find the volume of the indicated region. The region that lies under the paraboloid z=x^2+y^2 and…

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Q: Find the volume generate

A: Given,      y=-x4+3x3,  y=0 revolving about yhe y-axis.

Q: Component A is bounded by the planes x=0, x=2, z=-y and z= = y²/2. (a) Sketch A and find its volume.

A: The region is bounded by the plane  x=0, x=2, z=-y, z=y22

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