8. Sketch the graphs of the functions f and g, and find the area of the region enclosed by these graphs and vertical line x=a and x=b. g(x)=x+2, f (x)= x² – 4; a =–1,6=2

Q: Q1. Find the area of the bounded region shown in Fig 1. [8] y = x 2 y = \x+2 -2 2 Fig 1

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Q: 1.) Find the area bounded by the region x = 4y + 4 and y= x+ 2

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Q: 14 12

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Q: 6x 3x The bounded region between the graphs of f(x) and g(x) has area 5 x2 +1

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Q: Find the area of the region. f(x) = (x – 5)3 %3D g(x) = x – 5 y 1.5F 1.0 0.5 1 2 4 /5 6. -0.5 -1.0…

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Q: Find the are of the region under the graph of f on [a,b] f(x)=e-x/2 ; [-1,14]

A: Given, fx=e-x2,           -1≤x≤14 Since fx is continuous and positive on the interval -1,14, area…

Q: If R is the region between the graphs of the functions f(x) = 8x³ + 7x² + 28x + 6 and g(x) = 7x³ –…

A: Plot the curves f(x) and g(x) and find the region bounded by the curves over the interval [ -2,2].

Q: Find the area of the region enclosed between these graphs and the vertical lines x=3 and x=4.…

A: Here we used area between the two curves

Q: 1. Find the area of the region bounded by y = x2 + 4, y = x, x = 0, and x = 3.

A: The equation of curves are C1: y = x2+4C2: y = xC3: x = 0C4: x = 3 To evaluate: The area of the…

Q: To determine the area of the region bounded by x = y^2 - y - 6 and x = 2y + 4, first we have to…

A: find the area between curve

Q: Find the areas. of. the region k bounded by.the graphs of the given equations R .y=0,X=2. and. x=…

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Q: 1. Sketch the region R between the functions y = 2x and y = x2

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

Q: I. FIND THE AREA BOUNDED BY THE FF CURVES AND LINES: 1. The loop of y = x *(4 - x) 2. у %3 бх - х, y…

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Consider the following. f(x) = x(x² – 11x + 35) x2 g(x) %D (a) Use a graphing utility to graph the…

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Q: Find the area of the region bounded by the graph of f(x) = x3, the x-axis, and the vertical lines x…

A: We first sketch the region using a graphing calculator.

Q: 1. Determine the area bounded by curves y = -x² + 4x + 3 and y = −x³ + 7x² − 10x + 5 over the…

A: We have to find the area bounded by the curves.

Q: Find the area of the region between the graphs of f(x) = 3x + 8 and g(x) = x2 +2x +2 over [0, 2]…

A: according to graph area of shaded region can be written as given below A=∫02fx-gxdx

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 under the curve y a. f(x) = x³ [2,3] b. f(x) = 3√x [1,4] (x) over the stated…

A: a) fx=x3          2,3 b) fx=3x          1,4 Find the area under the curve y=fx over the stated…

Q: 1. Find the area of the given region bounded by the graphs of y, = x² + 2, y2 = 4 – x², and y3 = 2 –…

A: The area of the region enclosed by curves A=∫abfx-gxdx. Obtain the intersection points of y1,y3 as…

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

A: Given Graphs y=x2-3xy=x3+x2-12x First we find point of intersection of graphs…

Q: What is the area of br^2 of the region bounded by

A: The given question is taken from the calculus in which we have to find the area bounded by the given…

Q: Determine the area of the given region. 3. Region between f(x) = x - 4x and g(x) = x.

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Q: Which of the following represents the area of the region bounded between the graphs of y = +3z and y…

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Q: Find the area of the region bounded by the graph of a.) x = y² - 4 and x - 2y =1 b.) y = 9 - x²…

A: (a) Given: x=y2-4……1x-2y=1……2

Q: 7. Find the area of the region bounded above by the line y 3, below by the graph of y VT, between…

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Q: Consider the following functions. f(y) = y2 + 5 g(y) = 0 y = -1 y = 2 Sketch and shade the region…

A: We have to first sketch the given region.

Q: Find the area of the region bounded by the y-axis, the line x = 3, and the functions f(x) = 1/2 (x^2…

A: Consider the functions fx=12x2-3x+10 and gx=-13x-1 Find area of region bounded by the y-axis x=0 and…

Q: 1. Find the area bounded by y2 = 4a(y – x), y² = 2a(x + 2y – 3a) and the x-axis.

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Q: 1)Find the area bounded by the given curves. a)y = 3x2 − 12 and y = 0 b)y = x2 − 2 and y = 10 −…

A: we have been given the function y=3x2-12 and y=0 and the formula of the area under the curve is…

Q: 1 75. Find the area of the region bounded by the curve f(x) = (16 + x²)-3/2 and the x-axis on the…

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Q: A region bounded by f(x) = x² + 1, y = 1, and x = 2 is shown below.

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Q: y = x2-4r + 5 y y=x +5

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Q: 9-1. Find the total area of the regions enclosed by the given curves: (a) x = y2 and x = y+2 (d) y =…

A: First find out the intersection points of both the curves than integrate in that interval with…

Q: 2 3. Find the area of the unbounded region between the x-axis and y=- for x26. (x-4)

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Q: 1. Determine the area of f (x) = 3 + 2x – x² and above the x axis. - X - 3x 3 3 2. Determine the…

A: To find the area of f(x) and above the x-axis, use the formula, A=∫-13f(x)dx and substitute f(x)…

Q: Find the area of the region bounded by the graphs of f (x) = x³ and g (x) = x. (A) 1 c) 0.5 3.

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Q: Consider a graph of f(x)=x3−2x2+2 and f(x)=x. Find the area of the region bounded between the curves…

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Q: 1. Find the area bounded by the curves y = 3x'+1 x=0 y= 3x+1 x-1 А. 3/2 5 В. 3/4 2 С. 1/4 -2 -1/0 21…

A: y = 3x3+1 , y = 3x + 1

Q: Estimate the value of k (0 < k < 1) so that the region enclosed by y = 1/√1−x2, y = x, x = 0,…

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Q: Find the area of the region bounded by f(x)=-x^2+1, g(x)=2x+4, x=-1, and x=-2.

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Q: (4, 16) 10 Find the area of the region bounded by the graphs of y = 4x y = x2 and y = 4x. y = x2…

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Q: The graphs of the functions f (x)= /x– 4, g(x)=-x+2, and y=2 are shown in the figure. y \3+ f(x) =…

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Q: 2.) Verify the conclusion of Green's Theorem given F(x,y) = (ye",e") and C is the boundary of the…

A: The objective is to verify the Greens theorem for the given function.  Let C be the positively…

Q: 1. Find the area of the bounded region by x- y = 2, x= y73 and y = 8. Round your final result to the…

A: The given function is bounded by the x-y=2,x=y23 and y=8. Write the given equation as a function of…

Q: Find the are of the region bounded by the graphs of the given equations f(1) = 5x² and h(x) =r² + 1.

A: To find: The area of the region bounded by curve fx=5x2 and hx=x2+1. Formula used: The area of the…

Q: Consider the region, R, bounded above by f(x)=x5^2−6x+9 and g(x)=−3x+27 and bounded below by the…

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Q: 1. Find the area of the region bounded by y = x² - 3 and y = 5 - x².

A: This question can be solved using the concept of area bounded by two curves.

Q: 1-Find the area of the region R under y=x2 + 3x + 2 Between x=-1 and x=2

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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.