a) Find a vector parametric equation for the ellipse that lies on the plane y – 4x + z = -8 and inside the cylinder x2 + y? = 4. r(u, v) = for 0 < u <2 and 0 < v < 2n. ru (e) dA = ||d|| = . x, = usqrt3 (d) Set up and evaluate a double integral for the surface area of the ellipse. Surface area =

Q: A hole is drilled through the center of a sphere x²+y² +z? = R,², leaving a ring-shape solid. If the…

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Q: Find a parametrization for the cylinder x2 + y2 = 1.

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Q: A hole is drilled through the center of a sphere x² + y² + z² = R,², leaving a ring-shape solid. If…

A: A per your request , (b) The equation of sphere is given as , x2+y2+z2=Rs2 and Radius of drill=Rd…

Q: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

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Q: B- Find the area of the ellipse resulting from the intersection of the plane x+y+z=6 with the…

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Q: Jse Green's theorem to evaluate line integral | V8 + x3 dx + 9xy dy where C is a triangle with…

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Q: Let P = x2 – y², Q = -2ry and C denote the ellipse centered at (0,0) and passing through the points…

A: We will use parametric equation of ellipse to find line integral . As only one question allowed to…

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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: Find a parametric representation for the part of the cylinder y2 + z² = 36 that lies between the…

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Q: As shown in the picture at right, a solid is bounded below by the xy-plane, above by the sphere x² +…

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Q: Find the line integral with respect to arc length (a) Find a vector parametric equation 7(t) for the…

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Q: Find a parametrization of the hyper-boloid of two sheets (z2/c2)-(x2/a2)-(y2/b2)=1

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Q: Use Stokes's theorem to evaluate OF.dr for the given F and C. (c) F(x, y, z) = xzi+ xyj– 3xz k, C :…

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Q: Find the centroid of the portion of the sphere x2 + y2 + z2 = a2 that lies in the first octant.

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Q: 2. Show that the surface area of the solid of revolution obtained by revolving the curve y = 1/x, x…

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Q: 4. Consider the surface defined by the portion of the cone z = -Vx2 + y? that lies below the…

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Q: Given the scalar function f(x,y)=2xy,calculate the line integral along the ellipse curve from point…

A: Given data:- The given curve is, x29+y24=1 The given scaler function is, f(x,y)=2xy t E 0,π/2

Q: 1. Use a parametrization to express the area of the given surface as a double integral and then…

A: Here we will cylindrical co ordinate system to parametrize the surface  x=r cosθ , y = r sinθ , z=z

Q: Suppose that C is the curve of interesection between the plane z = x and the cylinder x² + y? be the…

A: Consider the given function. z=x and x2+y2=1 As, the function S be the interior of this ellipse…

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Q: Suppose F(r, y) = (e", e") and C is the portion of the ellipse centered at the origin from the point…

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Q: 28. Show that the ellipsoid 2x² + 3y² + z² = 9 and the sphere %3D x? + y? + z? - 6x – 8y – 8z + 24 =…

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Q: Use Green's Theorem to evaluate the line integral. y dx + 7x dy C: square with vertices (0, 0), (0,…

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Q: Consider the spherical surface E of center at C= (0,0.2) and radius 3. There are two planes parallel…

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Q: 2/ Find the line integrals of the ellipse+3= ellipse by x = acos(t) ,y= bsin(t) 1 , parameterize the…

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Q: Find a parametrization for the portion of the sphere x2 +y2 + z2 - 16 between the planes z =- 2/2…

A: we have been given a sphere x2+y2+z2=16 between the planes z=-22 and z=2

Q: Given the scalar functionf (x, y) = 2xy, calculate the line integral along the ellipse curve ²² +²²=…

A: We have to find the line integral along the given curve.

Q: 6. Use Green's Theorem to evaluate the line integral fc 2xy dx + (x² + y²) dy, if C is the ellipse…

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Q: # 4. Use Green's Theorem to evaluate the line integral P (y*dx +2xy° dy), where C is the positively…

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A: the correct answer is option: D .) Hyperboloid of 2 sheets

Q: Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c&gt;0. Determine a sufficient condition on c so…

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A: Given - Consider a uniform solid rectangular block of mass m and dimension 2a×2b×2c. To find - Find…

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Q: (10) Sketch the solid bounded by z = x2 + y? and z 4- x2 - y². Use cylindrical coordinates to find…

A: Given ∫∫∫Dx2+y2dVand solid bounded by z=x2+y2 and z=4-x2-y2

Q: Verify Green's Theorem by using a computer algebra system to evaluate both the line integral and the…

A: According to question, F = P i + Q j,   P(x, y) = 2x,   Q (x, y) = x3y6 .

Q: (a) The part of the surface z = 1+ 3x + 2y² that lies above the triangle with vertices (0,0), (0,…

A: a) Given part of surface is  z=1+3x+2y2, which lies above the triangle with vertices (0, 0), (0, 1),…

Q: Let C CR be the curve from (0,-7,0) to (-2,-3,-8) along the intersection of the surfaces x- y 7 and…

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Q: The intersection of the ellipsoid x2 + 4y2 + z2 = 4 with the plane z= 3x gives an ellipse C. A)…

A: Given that the ellipse C is the intersection of the ellipsoid x2+4y2+z2=4 with the plane z=3x.

Q: Suppose that C is the curve of interesection between the plane z = x and the cylinder x² + y? 1 (an…

A: Given that F(x,y,z)=xi+zj+2yk    ….(1) The equation of cylinder and plane is x2+y2=1  and z=x…

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: