5. Given the equations, y = 1+ Vx, x = -8, x = 8, and y = 0, answer the following questions: a. Sketch, or provide a screenshot from Desmos, with all of the equations on a single coordinate plane. In the sketch or screenshot, shade in the region that is enclosed by all equations. b. Write the definite integral that represents the enclosed area. Then, solve the definite integral using the Fundamental Theorem of Calculus. c. Explain, in words, why your answer in part (b) makes sense according to the graph you provided in part (a).

Q: First make a substitution and then ure integration by parts to evaluate the integral. 503 cos(02) de…

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Q: 1 S. cot ³ x dx

A: I have stated the used formulas in the solution

Q: (b) Use the Fundamental Theorem of Calculus to evaluate the integral .8 So da dx x-6 x+8

A: Need to use the fundamental theorem of calculus to solve this integral.

Q: Use any method to evaluate the integral. S 300 dw w?/100 - w? ... S- 300 dw w/100 – w? (Type an…

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Q: Evaluate the integral. 4 dx +C (1 – x2)ž NOTE: Enter the exact answer, without any trigonometric…

A: This question can be solved by the concept of integration by substitution(without any trigonometric…

Q: Set up and simplify, but do not evaluate, a definite integral (in terms of the parameter t) that…

A: The area between two curves can be calculated using the definite integral. The area between the…

Q: Decide on what substitution to use, and then evaluate the given integral using a substitution. (Use…

A: ∫xe6x2-4dx

Q: Write the definition of line integral and apply it to solve the following integral: z² dz Such that…

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Q: Suggest a trigonometric substitution for each integral, and then write the new integral that results…

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Q: (b) Use residues to evaluate the following definite integral: de 6+5 sin 0

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Q: The instructions for the given integral have two parts, one for the trapezoidal rule and one for…

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Q: S)Use the Fundamental Theorem of Calculus to decide if the definite integral exists and either…

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Q: Use the calculus of residues to evaluate the following integral 27 1 de. 1+8 cos? 0

A: Given integral is ∫02π11+8cos2θdθ

Q: use a geometry formula to find the exact value of the definite integral  4x dx on the interval [0,b]

A: The given function is 4x with the interval [0,b].

Q: 3 12Vェ) da

A: We apply usual  integration formula

Q: Represent the length and area of each ellipse as definite integrals and use Simpson's rule with n =…

A: Given: r=11+0.9 cos(θ) Differentiate with respect to θ drdθ=0.9 sin(θ)(1+0.9 cos(θ))2 drdθ=0.9…

Q: Calculate the definite integral by referring to the figure with the indicated areas. dx y = f(x)…

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Q: Determine a definite integral that represents the area of the petal that points to the right from…

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Q: 10 Sin h (Ux)

A: we have to find integration

Q: If you are going to evaluate the integral S V1 - 7w-dw using trigonometric substitution, the answer…

A: Since you have asked multiple questions in a single request we would be answering only the first…

Q: If t is in "minutes," g(t) is in "gallons/foot," and v(t) is in "feet/minute," then the integral…

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

Q: In evaluating the definite integral ff (x) dx, where f (x) = sinx, the angle should be in:

A: Given the definite integral ∫abf(x)dx where f(x)= sinx To determine : The angle part of f(x) is in…

Q: 1. a° cos(2)da = %3D

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Q: state the trigon metri e subs tition you would use to Find the indefinite integral Do Not integrate.…

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Q: [x- Cao+bo)]" Answer: e 4såsi (28å +2Si,)

A: We need to integrate w.r.t to y so put all other things other than y as constants. T=e-k-y2C

Q: e) Apply the Fundamental Theorem of Calculus in order to evaluate the definite integral: + 2)* dz =

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Q: Evaluate the indicated integrals if b is a positive real number constant. A. ftan () dx B. dx C.…

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Q: Set up a reference triangle and use it compute the given integral using trigonometric substitution…

A: Given that, In the above integral the term has , In that term numbers stuff is negative and x stuff…

Q: Consider cos 0 d0. What is the best-fit parabola to cos e using the (a) three points 69 = 0, 01 =…

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Q: Decide on what substitution to use, and then evaluate the given integral using a substitution. (Use…

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Q: 91. State whether the approximation MN or Ty is larger or smaller than the integral. (a) | sin x dx…

A: Since you have asked multiple questions, we can solve first question for you. If you want other…

Q: The graph off consists of line segments and a semicircle, as shown in the figure. Evaluate each…

A: As per the company guidelines we can solve first three  parts . I hope that the given solution will…

Q: SET UP the definite integral that will give the area of the shaded region (as shown below) using a.…

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Q: Decide on what substitution to use, and then evaluate the given integral using a substitution. (Use…

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Q: Question 12 Compute the following integrals: (a) fix√2x - Idr. (b) Sze-37 dz. Pay attention to…

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Q: Give the appropriate trig sub that would be used to evaluate dx. Do NOT Vx² – 16 solve the integral.

A: Given:     ∫x3x2-16dx

Q: into real and imaginary parts sech (x + iy)

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Q: Determine the integral solutions  fro which x and y are positive 2x + 5y = 17

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Q: Assume that x,y € R and z=x+iy € C. Calculate the following integrals using the given contour and…

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

Q: dz 3x2 +8

A:

Q: Work out the value of the definite integral (5x +2)°dx 1 [B – CP], where A, B, C,p are positive…

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Q: Find the following definite Integrals : Wir ( i+ sec?x)dx 2. a. X.

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Q: 03 cos(02) dO /2

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Q: Which of the following integrals will represent the area outside r = 3, but inside r = 6 sin(0)?

A: area = 12∫ab[outer2 - inner2] dθ

Q: By the even extension method applied to the function f(x)= e-16z, where a > 0, the value of the…

A: The given function is, f(x)=e-16x,x>0

Q: Find the value of the integral of g(z) around the circle |z – i| =2 in the positive sense when g(2)…

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Q: Given V9x? + 4 ax. Using a suitable trigonometric substitution, the resulting integral in terms of…

A: One of the two main parts of calculus which are concerned with the study of integrals, as opposed to…

Q: Evaluate the integrals .SiPi+7 j+(t+1)k] 4 S (sinti+(1+cost)j] -a4

A: In the given question, we have to find the integral of  ∫01[t3i+7j+(t+1)k] ∫-π4π4[sinti+(1+cost)j]

Q: sec{s)ds

A: We know, ∫abf(x) dx = ∫abf(a+b-x) dx

Q: y y= f(x) 12 18 24 6. 6.

A:

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.