find the Jolume of the salid genereted by revaling- the curve 4t'qy=36 between y = 0. and g=2 Gbout y-bis

Q: Find the differential euqation representing the family of cicles of radius unity and center anywhere…

A: The order of the  differential equation of a given relation is the number of arbitrary constants in…

Q: Find the center of mass of a thin plate of density δ = 3 bounded by the lines x = 0, y = x, and the…

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Q: Find the moment of inertia of the region between the parabola y=4-x^2 and the line y=1-2x from x=0…

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Q: Find the equation of the tangent plane to the surface y = 2x2 - z2 at the point (0, –1, –1) %3D | |

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Q: Q6/ Find the moment of inertia and radius of gyration about the x-axis of a thin plate bounded by…

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Q: Find the area of the surface obtained by rotating the curve }=2,3 from z = 0 to z = 1 about the…

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Q: . Sketch the curve parametrized by r(t) = (ltl + t, ltl - t)

A: The given curve is rt=t+t, t-t. Sketch the graph of the above curve as shown below.

Q: 2. Find the area of surface that is generated by revolving the portion of the curve x = V16 – y , 0…

A: Given, x= √(16-y) Therefore, f(y)= √(16-y) Differentiating the above equation with respect to y we…

Q: Find the center of mass of the lamina bounded by the graphs of the equations x = 64-y² and x = 0 for…

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Q: Find the points on the surface 6x² + y² – 3z² = 4 where the tangent plane to the surface is parallel…

A: We need to find points of intersection.

Q: Find equations for the tangent plane and the normal line at point Po (X0,Yo,Zo)(1,5,0) on the…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Show that the line normal to the surface xy + z = 2 at the point (1, 1, 1) passes through the…

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Q: Find the derivative dydx of the hyperbolic spiral r=aθ.

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Q: Find the points on the surface ry + yz + zx – x - 22 = 0 where the tangent plane is parallel to…

A: Given surface is  Fx, y, z=xy+yz+zx-x-z2 and we have to find the points on this surface where the…

Q: Find the surface area of the solid obtained when the parametric curve x = −2cost; y=2sint for 0 ≤ t…

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Q: Find an equation of the plane tangent to the surface sin(xyz) = at the point (T, 1,-)

A: sinxyz=12 at the point π, 1, 16. Consider, sinxyz=12⇒xyz=sin-112⇒xyz=π6⇒z=π6xy Differentiating…

Q: Find the center of mass of a thin plate covering the region between the x-axis and the curve y =…

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Q: Find the equation on the plane tangent to the following surfaces at a given point: z=exy; (2,0,1)…

A: The equation on the plane tangent to the surface z = f(x, y) at a given point (a, b, c) is given by:…

Q: 5. Find the length of the curve given by y= Vt* – 1 dt and 1<x<2.

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Q: Suppose that a semicircular wire has the equation y= V25- x and that its mass density is 8 = b- y.…

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Q: Determine the equation of the normal line to the surface z2 cos(x + 2y) at the point (2, –1, 0).

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Q: 12. Find the area of the surface generated when the curve y = ,on [1,2] is rotated about the x-axis.

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Q: Find the points on the surface ry + yz + zr - r - z = 0 where the tangent (5) plane is parallel to…

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Q: Find a parametrization of the circle of radius 6 in the XY-plane, centered at the origin, oriented…

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Q: 9. Determine the position of the centroid of a sheet of metal formed by the curve y = 4x - x² which…

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Q: Find the center of curvature of the curve y-Inx at (1, 0).

A: We Know that Centre of curvature for y=f(x) is κ=|f′′(x)|/(1+f′(x)3/2  -----(1)

Q: Find an equation of the tangent plane and a parametric description of the normal line to the surface…

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Q: 4. Find the area of the surface generated by revolving the curve y = v9 – x², - 2 < x < 2, about the…

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Q: Find the moments of inertia about the coordinate axes of a thin rectangular plate of constant…

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Q: Suppose that a thin metal plate of area A and constant density doccupies a region R in the xy-plane,…

A: A thin metal plate of area A and constant density δ occupies a region R in the xy-plane.

Q: A point moves along the intersection of the elliptic paraboloid z = 9x + 4y and the plane y = 4. At…

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Q: ) Find the points on the ellipsoid 6x² + y² + 3z² = 4 where the tangent plane is parallel to the…

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Q: Find rectifying plane of et, é, t at the curve t%3DL

A: Given curve is αt=et,e-t,2t.  To find the rectifying plane of the curve . Result: Let rt→=xt,yt,zt…

Q: Find the area of the surface generated by revolving the curve y =,0sx<2, about the x-axis.

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Q: 6. Find the area of the surface formed by revolving about the curve defined by y = x2 - x2, 1<x<3…

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Q: Does the the paraboloid z = 1 – 2x² - - y² have a tangent plane at some – x + y + z = 1?

A: Introduction: Let P0=(x0,y0) is a point in the domain of f, and S is a surface defined by a…

Q: Find the gardientoff at the indicatedpoint. %3D

A: Given,

Q: The area under the curve 4y2 =" (x2 + 1)3 was rotated about a line located to the left of y – axis…

A: The surface area of the solid can be obtained using the Pappus theorem. The surface area is…

Q: The dimensions of a closed rectangular box are measured as 89 cm, 61 cm, and 35 cm, respectively,…

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Q: At what point on the ellipsoid 2x^2 + y^2 + z^2 = 1 is the tangent plane parallel to the plane x + y…

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Q: Find the Surface avea Gevereatel by vevoluing the curve y-2r coso about te x-axis X-Sine

A: According to the given information, it is required to find the surface area generated by revolving…

Q: Find the equation of tangent plane to the surface z = ln (3x²) + 2y at A (1,0, In (3)).

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Q: 12. Find the area of the surface generated when the curve y =+,on [1,2] is rotated about the x-axis.

A: solution is given below...

Q: Compute the equation of the tangent plane to the surface: y = 1 - x2 - 2z2 At the point (0, 1, 0)

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Q: Find the surface of revolution of the curve y = 4 + 3x² that is rotated about the y- axis between 1…

A: We have to find the area of the surface of revolution of the curve y=4+3x2 that is rotated about the…

Q: Calculate S. xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to (1,3).

A: We have to calculate the integral ∫Cxydx+3y4dy Where C is the curve in the xy plane, x=y2 from 0,0…

Q: Determine the equation of the normal line to the surface z2 = cos(x + 2y) at the point (2, –1,0).

A: We have  z2 = cos(x+2y) ⇒cos(x+2y) - z2 = 0  So, F(x,y,z) = cos(x+2y) - z2  Thus,  ∇F(x,y,z) =…

Q: 2. Find the area of the surface obtained by rotating the curve: y = Vx +1,1<x<5 about the Х-ахis

A:

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: