Find the TOTAL geometric area bounded by the curve given by f(x) = x5 – x3 and the x-axis on the interval [–1, 1]. Sketch the graph of the function and shade the area you found.

Q: Find the area under the graph of the function f (x) = xe-* between x 0 and x = 5.

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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 of the region under the graph of the function f on the interval [1, 4], f(x) = 6Vx 32…

A: We have to find area under the graph.

Q: Find the area bounded by the line f(x) = 2x + 9, the curve g(x) = x² 4x + 14, and the two vertical…

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Q: Find the area between f(x)=x²-7 and the x-axis from x = 0 and x = 6. ...

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Q: Find the total area between the graph of f(x)=x2−5 and the​ x-axis from x=0 to x=6.

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Q: Find the area of the region under the graph of the function f on the interval [4, 5]. f(x) = 2…

A: The given function is fx=4x2 on the interval 4,5.

Q: 3. Find the exact area between the curve f (x) = -2x² + 12x + 7, the x-axis, and x=1 and x=5.

A: As mentioned, use the sigma formula to find the area A. Plot the curves. Divide the area between the…

Q: Find the area of the region under the graph of the function f on the interval [4, 11]. f(x) = 2x - 2…

A: use integral to evaluate the are bounded by curve over the interval

Q: Find the area under the curve y = x3 + 3x2 and the x-axis between x = 1, and x = 3

A: Find the area under the curve y = x3 + 3x2 and the x-axis between x = 1, and x = 3.

Q: Find the area of the region under the graph of the function f on the interval [5, 11]. f(x) = 4x - 6…

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Q: Find the area between the graph of x = and the y-axis on the interval [1, 4]. 4 3 2- 1 0.5 05 1 1.5…

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Q: Find the area of the region bounded by the graph of f and the x-axis on the given interval.…

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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 total area bounded by the x-axis and the curve f(x) = x2 + 3x - 4 over the interval [0, 4]…

A: The given curve is,

Q: Find the area under the graph of f(x) = 3x2, above the x-axis, and between x = 0 and x = 2.

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Q: Find the exact area between the curve f (x) = 9 – x², the x-axis, and x=-3 and x=3.

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Q: Find the area of the region under the graph of the function f on the interval [0, 9]. f(x) = 9x -…

A: given  f(x)=ax-x2Requirid area , A= ∫0qf(x) dx A=∫0q(qx-x2) dx  = ∫0qqx - ∫0qx2dx

Q: Find the total area bounded by the x-axis and the curve y = f(x) on the indicated interval. f(x) =…

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Q: Find the area bounded by the curve y = x² and the x-axis and the ordinates x = 1 and x = 3

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Q: Find the total area between the curve of the function f(x)=−4x+8 and the x-axis over the interval…

A: To find the total area between the curve of the function f(x)=−4x+8 and the x-axis over the interval…

Q: Find the area of the region under the graph of the function f on the interval [1, 4]. f(x) = 6Vx…

A: Given function is - f(x)=6x We need to find area of the region  under the graph of f(x) on interval…

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: Find the area of the region under the graph of the function f on the interval [1, 5]. f(x) = 3x³…

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Q: Find the area bounded by the line f(x) = 3x – 5, the curve g(x) = x² – 10x + 7, and the two vertical…

A: Area between two curves :   If f(x) and g(x) be the two continuous function on a,b and f(x) ≥g(x) on…

Q: Find the area under the curve f(x) =x+3x+7, above the X axis from x=4 to x=6.

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Q: f (x) = x – 2x – 3

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Q: Find the total area between the curve of the function f(x)=2x+4 and the x-axis over the interval…

A: We have to find the total area under f(x)=2x+4 and the x-axis over the interval [−9,−1]Recall that…

Q: The area between the graph of f (x) = x4 and the x-axis over the interval [0, 1]

A: Given equation - f(x) = x4Given intervals = 0, 1Let's draw the curve for the function given -  x…

Q: Find the area of the region between the curve and the​ x-axis. ​f(x)=36−x2​, from −8 to 8.

A: The function is given by fx = 36-x2;  -8≤x≤8 To evaluate : The area of the region between the curve…

Q: Find the area of the region under the graph of the function f on the interval [1, 6]. f(x) : square…

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Q: Find the area (in square units) of the region under the graph of the function fon the interval [-1,…

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Q: Find the area of the region under the graph of the function f on the interval [1, 4]. f(x) = 8/x…

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Q: Let f(x) = -a² + 6æ – 5. Estimate the area below the curve y = f(x) and above the x-axis using 100…

A: Let f(x)=-x2+6x-5 To estimate the area below the curve y = f(x) and above the x-axis using 100…

Q: Given the graph of the function f(x) below, use a midpoint approximation with 3 rectangles to…

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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: Find the area between the curves, f(x) = ;x² and g(x) : - x over the interval [0, 2] 10 3 - 1/3 2.

A: We first sketch the region

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: 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: Find the area A of the region between the graph of f and the x axis on [-1, 1].

A: Please see the white board for equation formulating the calculation of A.

Q: find the area between the curves: f(x) = x2  + 1,   g(x) = 3 - x   on the interval [0,3]

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

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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 total area between the curve of the function f(x)=−2x and the x-axis over the interval…

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Q: 1. Find the exact area between the curve f (x) = 9 – x², the x-axis, and x=-3 and x=3.

A: As mentioned, use the sigma formula to find the area A. Plot the curves. Divide the area between the…

Q: Find the area R- region in the first quadrant that lies below the curve y=x^2+4 and is bounded by…

A: The given problem is to find the area of region R bounded below by the given curve and the line.…

Q: Find the area bounded by the line f(x) = 3x - 5, the curve g(x) = x² – 10x + 7, and the two vertical…

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Q: Find the area between the curves f(x) = x^2 - 4x +12 and g(x) = x^2 - 2x + 2  for 0 ≤ x ≤ 3

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Q: Find the area between the curve y = x = 0 and x = 5. x(x 3) and the ordinates - LA B 3

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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.