17. Consider the quarter-circle of radius 1 and right triangles ABE and ACD given in the accompanying figure. Use standard area formulas to conclude that 1 sin e cos 0 1 sin 0 2 cos 0

Q: that Find flz)= r°cos20 + ir?sin po Such %3D is analytic-

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Q: Evaluate the integral 7 3i sinh dz, 2i)4 where C is the circle |2 – 2i| = 4 traversed once…

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Q: Find the length of the arc y = In(cos x) between the points (0, 0) and (pi/3, -In2).

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Q: 4. The area which is inside the circle r = 8 cos e and outside the circle r = 2 cos 8.

A: Introduction: In polar coordinate, the equation of a circle centered at the origin with radius a…

Q: 25. What is the area of one of the petals of the rose curve r = 2 cos (20)? /4 /4 L con (20) do 8/"…

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Q: 2ーてスース+スニ。 A ind yh ans: スi:z )スてニリ人3:)

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Q: Given the integral csc²u=cot²u+1 1+ cos2u 2 1-cos2u 2 sec²u=1+tan²u B cos²u=- sec (x-1) tan4…

A: According to our guidelines we can answer only one question or three subparts of a question. Other…

Q: Given: In (sin² 3t) = e = arc cot (w) Find: dw dt O(1+w²) (6cot (36) e) [ Oet+w²et - 6tan (3t) - 6w²…

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Q: 2. Consider a circle centered at the origin with radius r. Use trigonometric functions and…

A: Explanation of the answer is as follows

Q: Consider the quarter-circle of radius 1 and right triangles ABE and ACD given in the accompanying…

A: Here the objective is to prove the identity given below:

Q: 1₁ -1.2 - St 0 √x sin ex dx

A: Given :   I1=∫01.2xsinex3dx

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Q: If I, = e cos" zdr, where n > 2, Prove that i) I, =1-n e sin z cos"- rdr

A: Given,  In=∫0π2e-xcosnxdx solving by using Integration by parts, Formula of integration by parts,…

Q: 4. Write down the name of the curves and shade the region that is inside of r = 2 + 2 sine and…

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Q: 7. two circles with areas 2T cm2 and 200T cm2 uoluos ofra

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Q: O! Find tue indefinite integrel by q meleing tue febstelion d=4 Sin o. 2 *7 16-x"

A: To determine: by substituting x = 4sinθ

Q: 4. Find the area common to the circler=3 cos 0 and the cardiod r=1+ cos0.

A: Find points of intersection and solve the given problem. A detailed solution is given below.

Q: Consider the quarter-circle of radius 1 and right triangles ABE and ACD given in the accompanying…

A: From the given fig. Using standard area formulas, We need to show that…

Q: 2. SL intersect IR at M. SI 1 RI and LR IR. Prove ASIM ~ ALRM M.

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Q: Find the area lying outside r = 4 cos 0 and inside r = 2+2 cos 0.

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Q: du nd az and " if z = 5 tan-'(x)and x = e" + In(v) at tl-

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Q: 4.) Find the area inside the circle r=v2sin (0) and the lemniscate r2 = cos(20)

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Q: Find the areas of the regions . 1. Shared by the circles r = 2 cos u and r = 2 sin u 2. Shared by…

A: 1. Area of the region shared by the circles r = 2 cos u and r = 2 sin u Replace the variable u by θ…

Q: A- Using Simpson s Rule to estimate the area of the circle (r-4), c (1, 0) with (n=4). B- Prove…

A: The equation of a circle having center C(1,0) and radius r=4 is given by: x-12+(y-0)2=42…

Q: 2. ind the length of an arc of the cycloid given by, x = a(0 + sin0), y = a(1 + cos0), Ar

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Q: b) Find the common area between r= 4 cos (0) and %3D r=2 sin (0).

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Q: The area between the two loops of the limacon r = 1 + 2 sin 0.

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Q: 29. Area = %3D EF, FH, and HI are tangents. EI is a diameter. E 2ft.F J 8ft. H

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Q: 2. Find the length of the asteroid x = cos't and y=sin't , 0<t<2n. x = cos' t y = sin³ t

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Q: 2. Find the area enclosed by T = @ cos 30-

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Q: Look for the smaller loop of the limacon r = a(1+2 cos 0) and solve for its area.

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Q: What formula would be used to calculate the right rectangle approximation of " (sin x · cos x)dx…

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Q: As shown in figure 5.111 to the right if the two small semicircle, each of radius 1 unit with…

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Q: y = sin x 2

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Q: Inside the circle r =3 sin 0 and outside the cardioidr 1 + cos 0

A: The area that we have to calculate is the shaded area in the diagram above.

Q: The integral of the right hand member of the first order LDE A B C с D -2sin³0 3 COS 2 2sin³0 3 sin…

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Q: 1. Use Green's Theorem to evaluate [(3ye* -7 sin (3x*))dx + (5x* +2x*y* +8e*-y* )dy, where C is the…

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Q: 7. Find the area of the region that lies inside both of the circles; [Ans: }( – 1)] r = 2 sin 0, r =…

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Q: The area between the two loops of the limacon r = 1+2 sin 0.

A: Question- The area between the two loops of the limacon          r = 1 + 2  sin theta .

Q: Find the length of the entire perimeter of the region inside r = 6 sin 0 but outside r = 3.

A: Given the equation of the polar curve is, r=6sinθ Solving the equation r=6sinθ and r=3 for θ,…

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Q: .2π π 1 -εrocos? θ r dr sinθdθ φdφ x 32 πο X = ?

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Q: Here are the paths of sin and cos. Can we find the area of one of the areas that are closed in…

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Q: a) By expanding sin (2x + x) and using double angle formulae, show that sin 3x = 3 sin x- 4 sin'x.…

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Q: Find the length of the entire perimeter of the region inside 15 sin 0 but outside r = 4.

A: Concept Used: Arc length formula Arc length l=∫abr2+drdθ2dθ

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Q: Find the length of the entire perimeter of the region inside r = 6 sin 0 but outside r = 1.

A: Given that  r=6sintheta and r=1 To find the length of the entire perimeter of the region

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.