A) Find the area bounded by f(x) = x² – x and g(x) = x³ for 0 < x < 4. In your answer graph both functions and show the area between these curves for indicated interval.

Q: Find the exact area between f(x) = 2x and g(x) = 15 – x from the vertical axis to their point of…

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Q: 18.Find the area bounded by the graph y = 30x – 3x² and the x – axis. %3D

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Q: The area enclosed by the curve y² = x(1-x) is given by .

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Q: 1. Find the area between the curve f(x)=√x and g(x) = x³.

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Q: Find the area under the graph of f(x)= x2 and above the x-axis, between x = 0 and x = 3.

A: Given, f(x) = x2, we have to find the area under the graph f(x) and above x-axis between x = 0 and x…

Q: Determine the expression for the area under the curve f(x) on the closed interval [0, 3] with n…

A: Definition of the definite Integral

Q: 18. Use limits to find the area between the graph of the function and the x axis given by the…

A: I have stated the used formula in the solution

Q: Find the area under the curve y=f(x)=x^3 from x=1 to x=3 using inscribed and circumscribed…

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Q: The area bounded by y = x2 + 5 and the caxis from x = 0 to a = 5 is

A: given, and the x-axis from x = 0 to x = 5 we have to find area bounded by .

Q: (a) Estimate the area under the graph of f(x) = 25 – x from x = 0 to x = 5 using five approximating…

A: Consider the equation as shown below: Here, the function fx =25-x2 And the object is to find the…

Q: 2. Find the area bounded by the two curves y=x²-6x²+ 8x and y = x² - 4x.

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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: the area bounded by the curves y=2-x2 and y=x2 is

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Q: Find the area enclosed by the curves f(x)=39-x2 and g(x)=x2

A: Given curves

Q: 1. Find the area bounded between the curves: y = x+1, y = -2x + 7 and x = 0 A. With respect to the X…

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Q: Approximate the area under the curve of y = x² - 2x + 3 over the interval x = -1 to x = 3 using 4…

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Q: 2. Find the area bounded by the two curves y = x² - 6x²+ 8x and y = x² - 4x.

A: We have to compute the area between the given curves using the double integration method by first…

Q: Let f(x)=-2x^ 3 +6x^ 2 -x+1, Use L 4 to estimate the area under the curve or the interval [- 1, 1]…

A: L4 is the left endpoint of the expression with 4 approximating rectangles.

Q: Find the exact area between the graphs of f (x) = 2x +1 and g (x) = 9 – x2. First find out where the…

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Q: Find the area between the curve  y = x2 + 3x + 2  and the x-axis from  x = −1 to x = 1.

A: We have to find the area between the curve y=x2+3x+2 and the x-axis from x=-1 to x=1.

Q: b. Find the area bounded by the curve y = Vxe-Vx and the x-axis from x = 0 to x = 4. (45 Ivarks)

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Q: Find the area bounded by the line f(x)=2x+2, the curve G(x)=X^2-6X +9 and the two vertical lines X=1…

A: Given:       fx=2x+2        gx=x2-6x+9        x=1,x=5 Area bounded by line and curve is:…

Q: 1. Find the area bounded between the curves: y = x°, x = 1, and y -8

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Q: Find the area bounded by the curve y=11+5x+x^2 and x-axis from x=−3 to x=−1

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Q: 3) Find the area bounded by the curves: x² – 4x + 1 = 3y and x2 + 2x + 2y – 5 = 0.

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Q: Find the area between the curve f(x)=x^3−110 and the​ x-axis from x=1 to x=3.

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Q: c) [ 1: Estimate the area under the graph of f(x) = x² - 1 from x = 1 to x = 3 using four…

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Q: 7. Find the area bounded by the curve y = x² + x + 4, the x-axis and the ordinates x = 1 and x = 3.

A: The area under the curve y=f(x) in the interval a, b is calculated using the formula: A=∫aby dx. We…

Q: 2. Find the area between the curve y = x3 – 6x2 + 8x and the x-axis.

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Q: 2. Find the total area under the curve f(x) = 2xe² from x = 0 and x = 5.

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Q: y = 9x²; x-axis

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Q: - Determine the area bounded by the curves y = -1² – 16x + 80 and y = (x – 4)°.

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Q: ii, Find the area under the graph y=5x4 between the limits x= 1 and x-3.

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Q: 3. Find the area bounded by the curve y = x² - 7x + 6 and the x-axis and the ordinates x 2 and x =…

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Q: 11. Find the area bounded by the curve y² + 2x - 2y - 3 = 0 and the y-axis.

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Q: Find the exact area between the curve f (x) = x³, the x-axis, and x=0 and x=4.

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Q: 7. Find the area bounded by the curve y = x e¯** , the x-axis and the maximum ordinate.

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Q: Find the area between the curves f(x) and g(x) 2. f(x) = 3x, g(x) = x² – 3x + 8

A: We will find the area between the two curves.

Q: c) [1 graph of : Estimate the area under the f(x) = x² + 1 from x = 0 to x = 2 using four…

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Q: Find the area under the graph of f(x) = r² + 5 between x = 0 and = 6. %3D

A: The function is given by fx = x2+5 To evaluate: The area under the graph of the function fx between…

Q: (12) Find the area bounded by the graphs of function y =x2 +1 and y = 2x -2 for -1 <x <2

A: Giveny=x2+1y=2x−2−1≤x≤2

Q: 12. Find the area bounded by the curve y = 3x(x - 2)² and the x axis.

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Q: Find the area bounded by the given curves. x2 = y - 1 and x = y - 3

A: From the given curve   x2=y-1 ; x=y-3   Now finding the point of intersection for both curves:…

Q: Find the area between the graph of y = x² – 2 and the x-axis, between x 0 and x = 4. %3D

A: Use integration to find the area.

Q: 1. Find the area bounded by the curves between the curve y=1+x and the line y-1 x

A: area between curve

Q: Solve. 1. Find the area of the region between the function f(x) = 4x + 1 and the x– axis on the…

A: Note :As per our guidelines we are supposed to answer only one question if there are multiple…

Q: 5. Find the area bounded by the parabola y = x^2 - 3 and the line y =1.

A: Given problem:- Find the area bounded by the parabola y = x2 - 3 and the line y = 1 .

Q: 4 Find the area bounded by the curve y = the x - axis, and the vertical lines x = -2 x²+4 and x = 2.

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Q: Find the area bounded by the curve y=10+4x+x^2 and x-axis from x=-3 to x=-1.

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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.