7. The area of region R bounded by the curve C is given by the line integral P-y dx + x dy 2 By using this result show that the area of an ellipse with the parametric equations x = a cos t, y = b sin t, 0 < t < 2n is nab.

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Q: The area of the surface obtained by revolving the curve y = x3 for 1<x<8 about the y-axis, is given…

A: Surface area obtaining by x=f(y) from x=a to x=b is about y-axis given in step 2.

Q: 4/ Find the area of the region enclosed between the curves y= cos 2x, y=0, x=TT/4, x= TT/2 ? ans/…

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Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0<t < 1 is…

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Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

Q: (a) Set up an integral for the length of the arc of the hyperbola xy = 1 from the point (1,1)  to…

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Q: 2) Show that the integral f. (cosyz dx – xz sinyz dy – xy sinyz dz) is path independent and find its…

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Q: Use Green's Theorem to evaluate the line integral. dx + (2 + x) dy -y2/2 - C: boundary of the region…

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Q: Use Green’s Theorem to evaluate  ∫ sin 5y dx + 5x dy  around the boundary curve C of the region R,…

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Q: 12. Consider the arc L : y = 1 + cos z from (0, 2) to (,1). Which of the following integrals give…

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Q: Find the area enclosed by the given parametric curve and the y-axis. sin²(t), y = 8 cos(t) X = y 0.2…

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Q: Find the area Inside the cardioid r=2 cos (6/3) and outside the circle of radius r = vZ centered at…

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Q: (b) Use Simpson's Rule with n = 10 to estimate the area of the surface obtained by ro the cosine…

A: I have solved b part as asked.

Q: 3) Find the integral with respect to u, for the arc length of the parametric curve x(t) = 1 + 3f,…

A: Since you have posted a question with multiple questions, we will answer the first one for you. If…

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: 1 1) To find the exact area of the surface obtained by rotating the curve y = 2, 0<x < 12 about the…

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Q: 6. Determine the volume V of the 3 dimensional lampshade formed (see diagram below) by rotating the…

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Q: 16. Compute the surface area of the surface obtained by revolving the given curve about the…

A: To compute the surface area of the surface obtained by revolving the given curve about the axis x =…

Q: The circle of radius 3 centered at the origin Ex. to calculate the area of the given region.

A: Given: The region between x-axis and the cycloid parametrized byr(t)=t-sin t, 1-cos t  for 0≤t≤2π

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 5y +…

A: Use Green Theorem to evaluate  ∫C5y+9exdx+10x+7cosy2dy where, C is the boundary of the region…

Q: Jse Green's theorem to evaluate the ne integral (2tan x _cos 2y³) dx + (7⁰s y + 2x (πº where is the…

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Q: (b) Set up a definite integral that is equal t length of the portion of the parabola y which serves…

A: Introduction: The washer method is used to calculate the volume enclosed by two functions. In this…

Q: 5) Evaluate the integral (e* - y) dx + (sin y + x)dy using Green's theorem where Cis bounded by y =…

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Q: Consider the polar curves C₂: r = 4 cos 0 and C3 : r = 4 sin 0. (a) Determine the slope of the…

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Q: Sc F. dr where F(x.y) = (3x² – 4v)i + (27 + cos(,³))j

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Q: 5. Evaluate the double integral (2x – y) dA where D is bounded by the circle with the origin as its…

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Q: 8) Let x= 2t - sin(2t) and y=- cos(2t). Set up an integral in parametric form to determine the…

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Q: Evaluate the line integral ſ. 8y² dx + z² dy + (100 – x²) dz over the quarter of the circle of…

A: To evaluate the below line integral. ∫c8y2dx+z2dy+100−x2dz Here c is the quarter of a circle of…

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. (3y +…

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Q: By using polar coordinates, evaluate the double integral -y²dx dy x² + y² over the region in the…

A: Answer is given using polar coordinates.

Q: 3) Show that the line integral is independent of path and then evaluate the integral: S. sin y dx +…

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Q: Use Green's Theorem to evaluate the line integral. (e-x²12-v) dx + (0-1²/2 + x) ay dy C: boundary of…

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Q: 3. By using Green's Theorem in the plane, evaluate | (y - sinx) dx + cos x dy where C is the…

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Q: Sketch the regions enclosed by the given curves. y = 4 cos(7x), y = 4 sin(14x), x = 0, x = T/14 y y…

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Q: Find the area of the region bounded by the parametric curve x = e³¹, y = e²¹ and the X- axis on the…

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Q: 1. Find the area of the surface generated by revolving the parametric curve x=r cos 0, y = r sin 0…

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Q: Evaluate |(2y+cos x)dx + (5x+ sin y)dy along the positively oriented curve C. C is the boundary of…

A: We have to evaluate the integral ∫2y+cosxdx+5x+sinydy along the positively oriented curve C. Where C…

Q: Compute the integral |(5y + pz2020, ) dx + (19x – e-4ycos(v) sin y) dy over the portion of the…

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Q: 8) Find the area of the surface generated by rotating about the x-axis the arc of the curve y x…

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Q: 10. Use a definite integral to find the exact area enclosed by the x-axis and the parametric curve =…

A: The given parametric curve is: x=2t3+1    and     y=2t-t2

Q: у 3 1 + sin х, 0 <x< т/2 y

A: The given curve is y=1+sinx and the limit is 0≤x≤π2 The Surface area is given by S=∫x2πy1+y'2dx or…

Q: Evaluate the line integral (4х + 5у)dx + (За — 6у)dy along the - curve C : x = 10 cos t, y = 20 sin…

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Q: 8. Calculate the area of the region within the ellipse x2/a² + y²/b² = 1 parameterized by x = a cos…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Consider the polar curves C2 : r = 4 cos 0 and C3 : r = 4 sin 0. (a) Determine the slope of the…

A:

Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0 <t < 1 is…

A: x=t3,y=t2      o≤t≤1    Given

Q: #3. Evaluate the line integral O (y dx- x dy) where C is the positively oriented circle with center…

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Q: ху — 4, 1 <x< 8

A: Surface area.

Q: Evaluate the integral S 4-y2 dx dy using polar coordinate

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Q: 1. The area under the curve with parametric equations z = 1 + t?, y = l(2 – t) in the interval 0sts1…

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Q: 3.4.6 Evaluate the line integral [sin y dx + (x- cos y) dy] taken in the anticlockwise sense, where…

A: Green's Theorem States that  Let C be the positively oriented, smooth, and simple closed curve in a…