Find the area of the region that lies inside the curve r = 2 + cos (2a), but outside the curve r = 2 + sin(a).

Q: Evaluate the integral / (2x + 3y)dx + (6x – y)dy along the parabolic arc y = x' from (0,0) to (1,1).…

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Q: Find the area of the surface obtained by rotating the given curve about the x-axis. - 10 10 cos (0),…

A: Find the derivatives with respect to θ

Q: Write an integral for the area of the surface generated by revolving the curve y = cos(4x) about the…

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Q: Use a double integral to find the area of the region. The region within both of the circles r = 6…

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Q: Find the area of the surface swept out by revolving the para- metric curve x = 5 cost + 5t sin t y =…

A: Area under curve.

Q: Which one of the integrals below represents the area of the region that lies inside the curve r = 3…

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Q: Sketch the region that lies between the curves y = 4 sin(x) and y = 3 cos(x) from x = 0 to x = 0.37.…

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Q: 1 dA, where R is the union of both regions bounded by the functions y = sin(3x) x2 + y2 + 4 and y =…

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Q: What is the area under one crest of the sine curve, from x = O to x = n? y=sin x dx 1/2

A: Given problem:-

Q: The area under the curve parametrized by x(t) = e", y(t) = 2et +1, where 0 St<1/3, is equal to.

A: As given, x(t) = e3t  y(t) = 2e3t + 1 where 0≤t≤13.

Q: Find the surface area of the figure generated by revolving x = 6 cos(t), y = 6 sin(t) where 0 <ts…

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Q: Find the area of the region outside r = 7+ 7 sin 0 , but inside r = 21 sin 0. Preview

A: Given Data The outside function is r=7+7 sin θ. The inside function is r=21 sin θ.   Equate the…

Q: If the area enclosed by the curve x = 4 cos³ t and y = 4 sin t (wheret e R) is A then what is A?

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Q: Find the area of the region outside r = 6+ 6 sin 0 , but inside r = 18 sin 0.

A: The given curves are,r=6+6sinθ=r1(let)and r=18sinθ=r2(let) We need to find the area of the sketched…

Q: Find the surface area of the object obtained by rotating y = sin(2x) , 0<x< about the x-axis.

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Q: Find the area enclosed by one loop of the lemniscate with equation r2 = 16 cos 20 shown in the…

A: Here, the given curve is: r2=16 cos (2θ) To find the area of one loop of the area bounded by the 2…

Q: Use a double integral to find the area of the region. The region within both of the circles  r = 4…

A: Find the intersection of the given circles as follows.

Q: 2. Evaluate the following integrals. (b) L, sin y² dy dx

A: Let us consider the double integral I=∫D∫f(x,y)dA, where D is the region of the integral.   The…

Q: Determine a definite integral that represents the arc length of r = 5 cos(0), 0 <0<= do 0. 2/4

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Q: Evaluate the integral / (2x + 6y)dx + (15x – y)dy a C arc y = x² from (0,0) to (1, 1). NOTE: Enter…

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Q: determine the area of the curve r = v V3 cos e + sin 0 that is below the r-axis.

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Q: 5) Determine the area of the surface obtained by rotating the arc y = Vx –1 around the x- axis from…

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Q: 4. Use Integration to find the exact area of the region common to r = 3 and r 6 cos(0).

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Q: Find the area of the region enclosed between y = 3 sin(x) and y = 3 cos(æ) from æ = 0 to a = 0.87.…

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Q: )Find the volume of the solid obtained by rotating the sine function between x 0 and x t about the…

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Q: Find the area of the region enclosed by the inner loop of the curve. r = 6 + 12 sin(0)

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Q: S sin y cos('/2) dy

A: ∫sinycosy2dy

Q: Evaluate the double integral y sin(2³)dA where R is the triangular region with the three vertices…

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Q: 2 sin 0 r dr d0 T/2 Jo

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Q: The area between the two curves in * the figure below is y = cos x y = sin 2x T/6 T/2

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Q: sketch the region of integration and evaluate the integral.

A: Given, ∫0π∫0sinxydydxwe have to evaluate the integral and sketch the region of integration.

Q: Sketch the region R whose area is given by the double integral. 9. 3 6. dy dx

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Q: The area between the two curves in the figure below is y = cos x y = sin 2x T/6 T/2

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Q: Find the area of the region that lies inside the curve r = 1 − cos theta and outside the curve r =…

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Q: Find the area of the region enclosed by the curve a = 6 cos(t) – 3 sin(2t), y = 7 sin(t) with 0 <t <…

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Q: Find the area of the closed region enclosed by the curve defined parametrically by x = 4 cos t, y =…

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Q: Determine a definite integral that represents the arc length of r = 6 cos(0), 0 <0< de 0. 下一2

A: Arc length

Q: Calculate the area between the curves r2 16 sin 20 and p2 16 cos 20 .Show by drawing. %3D %3D

A: We will draw region first, then find the area using double integral

Q: In the figure to the​ right, the equation of the solid curve is y equals StartFraction secant…

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Q: 1. Do both parts a) and b). a) Set up but do not evaluate an integral to find the arc length of the…

A: Solve the following

Q: The area of the surface generated by rotating the curve y = 1 = cos(3x), where 0 ≤ x ≤ , about the…

A: Let's find Area of surface,when rotated about x axis.

Q: Find the area under the curve y = sin 2t from t = 0 to t = π/4.

A: We have to find the area under the curve of a function y=sin2t , here we have the limit from 0 to…

Q: Find the area under one arch of the trochoid x 70-1 sin(0), y = 7–1 cos(0).

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Q: a The integral that represents the length of the curve 7 (t) = <2/t,cos (2t3/2), sin (2t3/2)) where…

A: Given curve is, r→(t)=&lt;2t, cos(2t32), sin(2t32)&gt;, t∈[0,1] Now,…

Q: b. Find by double integration the area enclosed by the curve r= a(1+ cos 0) between 0 = 0 and 0=T.…

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Q: Jy Find the area of the inner loop of the curve r = 9 cos 0 - 4 by using double integral.

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Q: - SxS Write an integral for the area of the surface generated by revolving the curve y = cos (3x)…

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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.