Find the area of the region bounded by the equations below. For each problem, clearly sketch the graph of the region, set up the appropriate integral(s), and evaluate the integral. y = x^2 + 1 and y = x^3 − 3x^2 + 1

Q: Sketch the region bounded by the graphs of the equations y = −x2 + 3x + 1, y = −x + 1 and find the…

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Q: Sketch the region bounded by the graphs of the equations and find the area of the region x = y2 + 1,…

A: Given Data The first equation is x=y²+1. The second equation is x=y+3.   Let, f(x)= y²+1 g(x)=y+3…

Q: Find the area of the region between the graphs of y = 16 – x and -1< x < 7. 3x 2 over the interval…

A: Given: The functions y=16-x2 and y=-3x-2, To determine: The area of region between the graph of…

Q: Find the area of the region bounded by the curves, x = -6, x = 6, the x-axis and the parabola x² +…

A: Graph:

Q: Find the area of the region bounded by the graphs of the equations                       y = x / (√x…

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Q: Find the area of the region bounded by the graphs of the equations. y = -x2 + 10x, y = 0

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Q: Find the area of the region bounded by the graphs of the equations.

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Q: Find by integration the area of the region bounded by the curves y=x2 and y=−x2+4x. Include a quick…

A: Use area under curve

Q: Sketch the region bounded by the graphs of the equations f(x) = x2 + 2x, g(x) = x + 2 and find the…

A: The region bounded by the graph is as below

Q: Sketch the region whose area is given by the definite integral. xp (16 – x| – 6)

A: The definite integral corresponds to the area bounded by the graph of f(x)=9-|x-9| and the x-axis…

Q: Find the area of the region bounded by the graph of y = x /(x2 + 1) the x-axis, and the line x = 3

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Q: Find the area of the region bounded by the graphs of the equations y=x−x3, x= 0, x= 1,and y= 0.

A: Given equations: y=x-x3x=0x=1y=0 The region bounded by the graphs is shown below:

Q: Find the area of the region bounded by the graphs of y2 = 4x and the line 2x - y = 12

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Q: 4. Find the area bounded by the curves y = 2x – x2 and the x-axis

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Q: y = 4Vx +. 1 y = 0, x = 1, and x = 9

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Q: Find the area whose region is bounded only by the graphs y + 1 = - x^2 + 7, and y - 3 = ( x - 2 )^2…

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Q: Find the area of the region in the first quadrant bounded by the line y=3​x, the line x=4​, the…

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Q: Find the area of the region bounded by the graphs of the given equations. y​ = 5 - x^2, y = 5 - 5x

A: Given that, Two equations are following, y = 5 - x2  and y = 5 - 5x We have to find the area bounded…

Q: Find the area of the region bounded by the graphs of the equation y = x and y= x².

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Q: Find the area of the region bounded by the graphs of the equations.

A: If there are two curves given,say, f(x),g(x) over an interval [a,b] such that over this interval…

Q: Find the area of the region bounded by the graphs of the equations y = 1+ Vx y = 0,x = 0, and x = 8

A: Solved below

Q: Determine the area bounded by the curves y = 6x − x2 and y = x2 − 2x and illustrate an accurate…

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Q: Sketch the region whose area is given by the definite integral. 8. x dx

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Q: Determine the area bounded by the curves y = -x² – 16x + 36 and y = (x – 2)².

A: Answer

Q: Find the area of the region bounded by the graphs of the equations.                  y = √x(1 − x),…

A: Given function is y=x1-x. Graph the function as shown below.

Q: Find the area bounded by the graphs of y = 3x2; x = 0; x = 2; and the x axis.

A: Given,

Q: Graph the equations and shade the area of the region between the curves y = x3 and y = x2 − 7x over…

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Q: Find the area of the region bounded by the graphs of the given equations. y =– x- 3' y = 0, x = 4, x…

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Q: The definite integral that calculates the area of the region enclosed by the curves y=4 - x² and y =…

A: Solution of the problem as follows

Q: Find the area of the region between the graphs of the equations = x and y = 2 – x.

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

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Q: Find the area between the curves:  and  y= -x2+4x+2 and y=x2

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Q: Find the exact area of the region bounded by y = -x² + 3x +1 and y = -x+1

A:

Q: Find the area of the region bounded by the curve y = 9 - x2 and the x-axis. Answer:

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Q: Find the area of the region bounded by the graphs of the functions y=9-x² and y=x+3.

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Q: Find the area of the region by bounded by the graphs of y=x2+2x and y=-x+4 on the interval [-4,2]

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Q: Use the rectangles in each graph to approximate the area of the region bounded by y = 5/x, y = 0, x…

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Q: Find the area of the region bounded by the graphs of y = (square root (x)) and y = x/2. Leave the…

A: Find the area of the region bounded by the graphs of y=x and y =x2

Q: Find the area of the region bounded by y = x³ - 8 the x-axis and the y axis.

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Q: find the area of the region bounded by the curves y=12x/(x^2+4) the x-axis x=1 and x=4

A: The region corresponding to the equations is given below, Consider an elemental vertical strip of…

Q: Find the area of the region above the x-axis bounded by the function y = 4x - x2 – 3

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Q: Find the area between the curves x2 and 6x – 5

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Q: Find the area of the region between the graphs of y = 13 – x² and y = over the interval 3 < x < 8. –…

A: Solve by integration

Q: find the area of the region bounded by y=(x+2)^2-1, the x-axis, x=-1, and x=2.

A:

Q: Solve using iterated integrals. (with graph) Find the smaller area cut from the circle x2 +y2 = 25…

A: The objective is to find the smaller area cut from the circle x2 +y2 = 25 and x=3.

Q: Find the area of the region bounded by the graphs of the equations.                 y = 2√x − x,…

A: Given: The equations, y=2x-x and y=0.

Q: Find the area of the region that is enclosed between y=4x^2-x^3+x and y=x^2+3x.

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Q: Find the area of the region in the second quadrant bounded by y= -x and y=x^4

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Q: Find the area of the region bounded by the graphs of the equations: y = sin x y = Cosx - This can be…

A: Given query is to find the area bounded by the region.

Q: Find the area of the region bounded by the graphs of the equations. y = -x2 + 10x, + 10x, y = 0

A: Clearly we can rewritten the equation y=- x2+10x  as                                        y-25= -…

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.