f(x)=6cos2x Find the area of the region enclosed by the tangent line of the function f(x) at x=2 an the coordinate axis.

Q: Find the area between the curve y = x² + 8x + 12 and the x-axis from x = -2 to x = 0. (Give an exact…

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Q: [3tan(x)² dx 2

A: Concept: Integration: The Sum of discrete data is denoted by Integration. The integral is calculated…

Q: Use integration by parts to find the indefinite integral ∫x√(x − 1) dx

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Q: Use the midpoint rule with n = 6 to estimate the value of the integral (x³ + x) dx.

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Q: Use integration by parts to find the indefinite integral.

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Q: Find the area between y=x+2 and y= x2−1. Round your limits of integration and answer to 2 decimal…

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Q: Find the area of the region between the curves y = sin(x), y = sin(2x), x = 0, and x = t/2. Area…

A: Given curve is : y=sinx   .......(1) y=sin2x   ......(2) To find the area of region between curve…

Q: Use integration by parts to find the indefinite integral ∫x e1-x dx

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Q: se the midpoint rule with n = 6 to estimate the value of the integral (x³ + x)dx.

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Q: Find the area A of the region that is bounded between the curve f(x) = -4 sin () and the curve g(x)…

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Q: Use the midpoint rule with n = 6 to estimate the value of the integral (x3 +x)dx

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Q: se a definite integral to find the area of the region between the curve y = 12x and the x-axis on…

A: Given query is to find the area under the curve for the function.

Q: find the relation error use the trapezoid rule to approximate the definite integral. Use n=4 xe* dx

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Q: Find the area enclosed by the curves f(x) = 33 - x² and g(x)=x². The area (rounded to two decimal…

A: The region bounded by the curves fx=33-x2 and gx=x2 The formula to calculate the area bounded by the…

Q: Find the survace area of revolution about the x-axis of y= 4sin(2x) over the interval x is greater…

A: Given: The curve y=4sin2x for 0≤x≤π2 To find: Surface area of revolution of given curve about…

Q: Find the area of the region enclosed by the tangent line of the function f(x) at x = 2 and the…

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Q: Find the area between the curves y = ln x and y = ln 2x from x = 1 to x = 5.

A: The given two curves are,

Q: Use the midpoint rule with n = 6 to estimate the value of the integral (x³ + x)dx

A: We need to use the mid-point rule to find the value of the definite integral.  The mid-point rule…

Q: Use the right-endpoint approximation to approximate the area under the curve of f (x) + 1 on the…

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Q: Use the Trapezoidal Rule to evaluate the integral [x-'dx. .-1 1

A: Equation is f(x)=1xThe value of the table for x and y x 1 1.25 1.5 1.75 2 y 1 0.8 0.6667 0.5714…

Q: sin(z)

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Q: integral from 0 to 1 of square root of (2x - x^2) dx

A: Given, ∫012x-x2dx  We have to evaluate it..

Q: (a) Use the Midpoint Rule with n = 6 to approximate integral 8 x-6 S8 dx 6 x+8 ?

A: As per the instruction, we will be answering the first question. Given:

Q: Area under the curve y2 = 4x from x=1 to x=4

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Q: Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y…

A: Consider the provided curve,  Bounded above by the curve y=ex2 and below by the curve y=e-x2. And on…

Q: Calculate the area confined by between the portion of the graph of the function f(x) = -(x – 2)² + 4…

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Q: find the relation error use the trapezoid rule to approximate the definite integral. Use n-4 xsinx…

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Q: Sketch the region bounded above by the curve (9e^−x)*sin(πx), the x-axis, and x = 1, and find the…

A: We have to sketch the region bounded by the curve 9e-xsinπx, x-axis, and x=1, and also find the area…

Q: f (x) = x² and g(x) = 3x – x2 Зх — 3. Find the area between the two curves by integration.

A: We have to find area

Q: find the relation error use the Simpson's rule to approximate the definite integral. Use n=4, , Inx…

A: Simpson' s rule for integration:…

Q: Find the area bounded by the curves y=sin x and y= cos x from x =0 tox= 7. Make a sketch and shade…

A: The region bounded by the curves y=sinx and y=cosx from x=0 to x=π

Q: Using the trapezoidal rule with n = 4, estimate the integral / (x (x+1)² dx.

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Q: integral from 0 to 1 of square root of (1- square root of x) dx

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Q: Find the area of the region that is bounded above by the curve f(x) = (x + 2)² and the line g(x) = 4…

A: NOTE: Refresh your page if you can't see any equations. . here we have first, we need to find the…

Q: Graph the function y = -0.3x² + 1.5x over the interval [0, 6]. (0, 6] by (-2.25, 2.25] a. Integrate…

A: Given function y = - 0.3x ^ 2 + 1.5x over the interval [0, 6].   To find  a. Integrate y = - 0.3x ^…

Q: Find the area of the region enclosed between f(x) = 0.2x + 6, g(x) = x, x = -2, and > Area =

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Q: 7. Find the area bounded by the curve y = x exp(-x²), the x-axis and the vertical line p through the…

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Q: Example: Find the area of the region in the first quadrant that is bounded above by Y1 =Vx and below…

A: The graph is shown below:

Q: Find the area under the curve y = lines x = 1 and x = 5. 3x² + 4x + 1 bounded by the x-axis and the

A: Area under the curve y=3x2+4x+1 bounded by x-axis and x=1 to x=5.

Q: 25 dx Evaluate by dividing the interval into 8 equal parts 1 x2 +6x+ 10 by using Trapezoidal rule.

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Q: Find the area bounded by y = 0.5(x- 2)² – 1 and the X-axis from x-3 to x-5. Show the sketch with…

A: y= 0.5(x-2)2-10 =0.5(x-2)2-11 = 0.5 (x-2)22 = (x-2)2x - 2 = ± 2x =2+2    x~ 3.414orx =2-2x ~ 0.586

Q: Find the area of a triangle bounded by the y axis, the line 2 f(x) = 6 - x, and the line…

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

A: To find the area:

Q: Find the area the region lying between the x axis and the curve y=(ln(8x)-1)/x2 to the right of the…

A: Given curve is y=ln(8x)-1x2 Since region lying between x-axis and the curve to the right of the line…

Q: Find the area of the segment cut off from the curve y = sinx, 0≤x≤nt by the line y = 1/√2. HT 1/2 y…

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Q: Find the area between the curve  y = x2 + 10x + 16  and the x-axis from  x = −2 to x = 1.  (Give…

A: To find the area between the curve y=x2+10x+16 and the x-axis from x=-2 to x=1. This is given by…

Q: Evaluate the integral. ( Remember to use absolute values where appropriate.  Use C for the constant…

A: The given integral ∫axx2-bxdx can be evaluated as, ∫axx2-bxdx=∫axxx-bdx…

Q: Find the area of the region bounded by the graph of f(x) = x sin x and the x-axis between x = 0 and…

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Q: Integrate using trigonometric substitution Integral of x/(x^2-1)dx

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Q: Approximate the area under the curve f(x)=sin2(πx) using the midpoint approximation rule with 3…

A: Using a sum of midpoint rectangles calculated with the given formula

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.