1. Let S be the surface generated by the curve z² + y +4= 0 when it is revolved about the r-axis. (a) Give a set of parametric equations for S. (b) Use the answer in (a) to find the normal line to S at (1, ).

Q: Find the parametric equations of normal line to the surface given by f(x,y) = tan Vx2 + y² at the…

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Q: EXAMPLE 9 Find the tangent plane to the surface with parametric equations x = u2, y = v2, z = u + 2v…

A: Given parametric equation x=u2 , y=v2 , z=u+2v

Q: Find an equation of the tangent plane to the given parametric surface at the specified point. x = u…

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Q: 12- Find an equation of the tangent plane to the given parametric surface at the specified point.…

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Q: 25) Show that the curve with parametric equations x = tcos(t) , y = tsin(t), lz = t ies on the cone…

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Q: Find equation of the plane tangent to the following surface at the given point. an z = tan(xy):…

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Q: Find parametric equations of the tangent line to the curve of intersection of paraboloid z = x² + y²…

A: We know that tangent line to ellipse will be perpendicular to normal of both the surface. Therefore,…

Q: Consider the surface x2 + y2 − 2xy − x + 3y − z = −4 and the point P(2, −3, 18). Determine the the…

A: Given surface is x2 +y2-2xy-x+3y-z=-4 and point is P(2,-3,18)

Q: 4. Find the parametric equations for the tangent line to the curve of intersection of the y? z1…

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Q: Find the equation of the line tangent to the parametric curve (pictured below) defined by I=, y=t+t…

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Q: Find parametric equations of the tangent line to the curve of intersectic paraboloid z = x² +y? and…

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Q: 2. Find a set of parametric equations for a particle that oscillates on the line y = 6x - 5 between…

A: formula:

Q: 4. Find an equation of the tangent plane to the surface z = xy+8 at the point (1,1,-10) xy-2x

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Q: 2. Find the parametric equations of the tangent line to the curve as result of the intersection of…

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Q: 3. Let f(x, y) = xy+ y². Find the equation of the tangent line in parametric form in the x-direction…

A: Solve the following

Q: 15. Find the radius of curvature at 0 = -n of the parametric equations x = 3(0 + cos 0), y = 3(1 –…

A: For the parametric curve x=x(θ), y=y(θ)Curvature K=|x'y''-y'x''|[(x')2+(y')2]32Radius of…

Q: Find an equation of the tangent plane to the given surface at the specified point. z = 2x2 + y2 -…

A: We know that equation of tangent plane to the surface z=f(x,y) at the point (a,b,c) is

Q: III. Find the parametric equations of normal line to the surface given by f(x, y) = tan 2 V + y? at…

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Q: A Cartesian equation for the parametric curve x=24sin?e, y=21cos20, n ses 31/2 is given by O…

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Q: 27. Show that the curve with parametric equations x = t cos t, y =t sin t, z = t lies on the cone z²…

A: To Show: i. Show that the curve with the parametric equationx=tcost, y=tsint, z=t  lies on the…

Q: (7.5.0)- Find an equation of the tangent plane to the surface of z = sin(x – 2y) at the point %3D

A: Given F(x, y, z)=z-sin(x-2y) ∂∂xF(x, y, z)=∂∂x(z-sin(x-2y))=-cos(x-2y)∂∂yF(x, y,…

Q: 4. Consider the surface S parametrized by R(u, v) = (uv, sin u, cos v). Find an equation for %3D the…

A: The given curve S is parametrized by: R→(u,v)=uv,sin u,cos v Now, to find the equation of the…

Q: A curve is defined as the parametric equations: 1 x = acose, y=-asin 0, 0<0< 2n dy where a is a…

A: on solving this we get

Q: 16. Sketch the parametric curve x = Vt, y= sin t, 0<t< 2n. (b) (a) 1.0 (c) 101 1.0 0.5 0.5 0.5 05 10…

A: The graph of the parametric curve x=√t, y=sint , 0≤t≤2π is given as below,

Q: z = In(x²y*) + / x² + y²

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Q: 1. Find an equation of the tangent plane to the surface of z = sin(x - 2y) at the point (7,2,0).

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Q: Find the exact value of x²y³z ds, where C is the curve with parametric equations x = e-t cos(5t), y…

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Q: 4. Find the parametric equation of the tangent line to r(t) = 4costi – 3tj %3D at Po(2, –n)

A: Given: To find the parametric equation of the given tangent line as follows, r(t) = 4 costi- 3tj…

Q: Given the surface z = In(5x2 + 2xy - 3y2 – 12). (a) Find the equation of the tangent plane and the…

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Q: 1. Consider a fence whose base is the path with parametric equations |x(t) = cos³ t, |y(t) = sin³t,…

A: Given, A fence whose base is the path with parametric equations xt=cos3tyt=sin3t for 0≤t≤π2. Let the…

Q: Find an equation of the tangent plane to the given surface at the specified point z = √√√xy, (3, 3,…

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Q: Find an equation of the tangent plane to the given parametric surface at the specified point. x = u…

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

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Q: 1. Evaluate the length of the curve defined by the parametric equations x = cost, y=sint, z…

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Q: Find an equation of the tangent plane to the given surface at the specified point. z = 4(x - 1)2 +…

A: We have equation of the form     z= f(x, y)  And we have to find equation of tangent plane to the…

Q: Find a standard equation for the tangent plane and parametric equations of the normal line at the…

A: Introduction: If F(x,y,z) is a differentiable surface at the point (x0,y0,z0), the tangent plane to…

Q: Find the parametric equation of the line tangent to the curve of intersection of the cylinder…

A:

Q: 4. Find the parametric equations for the tangent line to the curve of intersection of the y z 1…

A: Given:

Q: Use definition 13.7.1 to find the equation of the tangent plane and the parametric equations for the…

A: Let the given function is,

Q: Find the parametric equations of normal line to the surface given by f(x, y) = tan v x2 +y2 at the…

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Q: Let C be the curve of intersection of the spheres x 2 + y 2 + z2 = 3 and(x - 2)2 + (y - 2)2 + z2 =…

A: Let fx=x2+y2+z2-3=0 &amp; gx=x-22+y-22+z2-3=0 At P=(1, 1, 1) ∇f=2x, 2y, 2z∇fP1, 1, 1=2, 2, 2…

Q: 1-) Given the surface z = In(-2x2 + 5y2 – xy). Find the equation of the tangent plane and the…

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Q: 1. Consider the graph of r(t) = 2t i+ tj +t° k Determine parametric equations of the tangent line to…

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Q: Eliminate the parameter in x = -t +2t and y =t-4 and then identify the parametric curve and sketch…

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Q: A curve is defined by the parametric equations z (t) = at and y (t) = bt, where a and b are…

A: Given,  We have  xt = at   and yt = bt  ;  where a and b are constants.  We have to find the length…

Q: Find an equation of the tangent plane to the given surface at the specified point. = In(x - 3y),…

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Q: Find an equation of the tangent plane to the given surface at the specified point. 3.  z = 2x2 + y2…

A: Given that: z = 2x2 + y2 - 5y, (1, 2, -4)

Q: Find an equation of the tangent plane to the given surface at the specified point. z= e(x^2)-y2,…

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Q: 3. Find the exact length of the parametric curve x = sin-1t, y = In /1 – t2, 0<t<

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Q: 30. Find parametric equations for the tangent line to the curve of intersection of the cone z = x +…

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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: