2. Consider the solid generated when the shaded region shown below is revolved around the x-axis. (This region is part of the region in problem #1.) The curve is y = x³ – 5x² + 3x + 9 and the line is y = x + 1. а. Draw a picture of the solid and a representative slice. b. Cind the volume of your slice. Write the integral(s) you would use to find the volume of the solid. Do not c. evaluate your integral(s). y=x25x+3x+9 10 2. -5 -10

Q: 1. Find the total area enclosed between the curves y 27² and y = 1¹ - 21². It is usually a good idea…

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Q: 2. Find the area enclosed by the line x = y+ 1 and the parabola y - 2x + 6. All answers must be in…

A: The given problem is to find the area enclosed by the given line x=y+1 and the parabola y2=2x+6.

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A:

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Q: Find the of the region bounded by the parabolas y – 6x = -x² and y + 2x = x² , area

A:

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A:

Q: The area of region bounded by the parabola y2 = 4x and the line %3D y = 2x – 4 is

A: We have to find the area of the region bounded by the parabola y2=4x and the line y=2x-4 .

Q: 3. Sketch the region enclosed by the curves/lines x = y, y =÷, x = 3 and find its area.

A: Given equations: x=y……1y=1x……2x=3……3

Q: Given y = 3x –- 2x +1, y =-x² + 2x +1 and a straight line y = 0.5x ; where x>0, 1. Sketch the curves…

A: Given y=3x3-2x2+1  , y=-x2+2x+1 and a straight line y=0.5x. We need to : 1) Sketch the curves and…

Q: 7. Find the area of the region enclosed by the curves y=x²-2 and x = y.

A:

Q: Calculate the area of the smaller region (to the left of the parabola) bounded by (x+5) 2=2y the…

A: The area of the smaller region( to the left of the parabola) bounded by  the x-axis, and the tangent…

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A: The shaded region is our required area

Q: III. Let R be the region below enclosed by the curves 2 x = y + 2y, x + 5y = 0, and x = 5. Set up…

A:

Q: 1. Find the total area bounded by the curves y=x²-3x and y = x³ + x² −12x.

A: Given curves are y=x2-3x y=x3+x2-12x Take  y=fxHere,   fx=x2-3x y=gxHere,   gx= x3+x2-12x

Q: 2. Find the dimensions of the largest area rectangle that can be inscribed in the region bounded by…

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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.