Example: Find the length of the cardioid r = 1 + sin 0, 0 02T. Fund length from and multi ply yfwo 2 Cr -Tr I+2sin+siiptca - 2+ 2sne L = de T(2 2+2sine dD 2 T(2 lenth I+S\n s 2L 2( 8 S2+25in2-2 N2-2sinD de 2-25in J4-4Sin S 2- 2sin& dt let 72 h= 2-2sinD du -2Co9d 2 Cos8 de 2-25inD -212-2sin 2 v S-du 4 11 FIN

Q: 4. Find the surface area of the paraboloid S defined by the equation: (r, y, z) = (u cos v, usin v,…

A: We will find surface area by using formula.

Q: 2. Use Green's Theorem (and only Green's Theorem) to evaluate |(2? + y)dx + (x² – y³)dy, where C is…

A:

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), E (0,4,0), and (0,0,2)

A: The vertices of the tetrahedron are (0,0,0),(1,0,0),(0,4,0),(0,0,2).  x varies from x:0→1y varies…

Q: An artist has decided to finish their piece of artwork by balancing it on a fulcrum and putting it…

A:

Q: 13 - x² – y? at (3, 2, 0), then If the point (1, 1, k) lies on the tangent plane to the paraboloid z…

A: z=13-x2-y2 Partially differentiating with respect to x, we get zx=0-2x-0=-2x Partially…

Q: x y 11. Use Green's theorem to evaluate O In(1+ y) dx –- dy , where C is the triangle with vertices…

A:

Q: Evaluate the triple integral |// xy dV where E is the solid tetrahedon with vertices (0,0,0), (7, 0,…

A: Consider the following: Equation of plane passing through 7,0,0,0,5,0,0,0,10x7+y5+z10=1x ​ranges…

Q: (b) A = 2pza, + 3z sin ø a, – 4p cos p a̟ and S is the surface of the wedge 0 0 < ¢ < 45°, 0 <z < 5

A: Let us solve the b part step by step with given A value

Q: Verify Euler's formula for the solids below. (i)

A:

Q: A clamped cubic spline S for a finction fis defined by a(x - 3) + (x-3)²+b(x – 3)³ - 2(x-4)-5(x -…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: sketch the solid whose spherical coordi- nates (p, 4, 0) satisfy the given inequalities. 0<p<2/cos…

A:

Q: 6. Evaluate (a + y) da dy dz, where D is the tetrahedron with vertices (0,0,0), (1,0, 0), (0, 1, 0)…

A: Introduction: The triple integral is applicable to find the volume of a solid enclosed by given…

Q: 10. Use Green's Theorem to evaluate the integral fe y dx + 2xy³ dy, where C is the square with…

A: Introduction: By applying Green's theorem, one can determine line integral through surface integral.…

Q: A=LDU where D-) then the humber of special solubion s bo Ax =0 is equal a-1/b.e/c.2/d.camt be deubie…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Let R be the trapezoid with vertices (1,-1), (3,3), (3,-3) and (9,9). Give a change of variables to…

A: Given that ,  R be the trapezoid with vertices (1,-1) , (3,3) ,(3,-3) and (9,9) . Let , the vertices…

Q: If the density of the frame element (shown in the figure) is given as p(x, y)=b+cy where a-9, b-1.2…

A: Given  Density μx,y=b+cy Where a=9, b=1.2, and c=3.1

Q: Consider the cune C: (x-8)+ 4-7) When parametneing C, A tungent veotoar is i O a) r (0) = (8 +3…

A:

Q: Consider a ball B of R". The PDE S Au(x) = 0 Iu(x) = g(x) Væ e ƏB. in B Væ e aB. has a unique…

A: Consider a ball B of ℝn.  We are given the following PDE.          ∆u(x)=0    in Bu(x)=g(x)   ∀x∈∂B…

Q: Considering f(, Y, z) = z + x²y and S as part of the cylinder y² + z² = 25 that is between the plans…

A: It is given that surface S is the part of the cylinder y2+z2=25 lies between the planes x = 0 and x…

Q: 13) De termine the surface area of the spiral ramp parametrized by = u cosQu) i + usin(2v)5 +vk with…

A:

Q: Show that the general solution to the two-dimensional Laplace's Equation. et = 0 is (Acosh(tx)…

A: We have to show that the general solution to the two-dimensional Laplace's Equation…

Q: show that fGx) = C,ex +C,e* s a solution to y for all real conatanto C, +C2, and then find C,, C,…

A:

Q: Let T be the tetrahedron with vertices (3, 0, 0), (0, 3, 0), (0, 0, 5) and the origin. Write z) dV…

A:

Q: Evaluate the triple Integral. 3xyz dV, where T

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Q: Use spherical coordinates. Evaluate z dv, where E lies between the spheres x2 + y2 + z² = 1 and x2 +…

A: Using spherical  coordinate we can solve it very easily.

Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (4,0,0), (0,3,0), and (0,0,2)

A: we have to find triple integral

Q: The classification of az дх2 дхду ду dz, dz O A. Elliptic B. None of these О С. Нурerbolic D.…

A: Answer and explanation is given below

Q: By using Green's theorem, evaluate P(2y)dx+(4x)dy where C is the triangle ABD with the vertices…

A:

Q: dA, where R is the trapezoidal region Cos 5. Use appropriate change of variables to evaluate x + y,…

A:

Q: Consider the solid Q that is generated in the first octant by the surfaces S, : 2x + 3y + 3z = 6 S,:…

A: The region S2 is clearly a parabola y2 = 2-x and it is symmetrical in the x-axis.

Q: z dV, where E is the solid tetrahedron with vertices (0,0,0), (2,0,0), E Find (0,6,0), and (0,0,3)

A:

Q: 5/10 Determine the coordinates of the centroid of the shaded area. y * = ky? a b.

A: Given:

Q: If the density of the frame element (shown in the figure) is given as p(x, y)=b+cy where a=9, b=1.2…

A:

Q: Semiinfinite conducting planes o = 0 and o /6 are separated by an infinitesimal insu- lating gap as…

A:

Q: Evaluate the triple integral II. x* dV, where T' is the solid tetrahedron h vertices (0, 0, 0),…

A:

Q: number 44 and it says use example 4 so i attached that as well

A:

Q: Evaluate the triple integral.  (x2 + y2) dx dy dz; W is the pyramid with top vertex at (0, 0, 2)…

A: The given triple integral is,

Q: Evaluate the double integral I2 = || y'dzdy T where T is a triangle with vertices (-4, 0), (0, 1),…

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Q: 3. WHEN THE LOAD IS UNl FORMLY DISTrRIBUTED ADRIZONTALLY A SNSPENSION BRIDGE CABLE HANGS IN A…

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Q: Use Green's Theorem to evaluate x dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

A:

Q: 10 The double integral 2x dædy, where T is the triangle with vertices (0,0), (3,0) and (6,3), equals

A:

Q: Example 6: Evaluate ez dz dy dx E Where E is enclosed by the paraboloid Z = 3+ x? + y² , the…

A: The given problem is to evaluate the given triple integral of given function ez for which region E…

Q: Evaluate fff zz dV where E is the solid tetrahedron with vertices 0(0,0,0), A(1,0,1), B(0, 1, 1) and…

A: Given that E is the solid tetrahedron with vertices 0,0,0,1,0,1,0,1,1. Suppose, the equation of the…

Q: Evaluate the integral // (10 · x + 9 · y)dA, where R is the parallelogram with: R vertices (-3,6),…

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Q: EXAMPLE 6 Find the volume of the box (parallelepiped) determined by u = i+ 2j – k, v = -2i + 3k, and…

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Q: a. Find a parametrization for the hyperboloid of one sheet x2 + y2 - z2 = 1 in terms of the angle u…

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Q: Q: A) Show Ih at W= a b qbceIR] is a Sub SP ceaf 2x2 ma mces SPace

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Q: 2 Find the average height of the paraboloid z = x + y over the square 0<x<1, 0<ys1. The average…

A: Use formula of average height

Q: A solid lies above the cone z = V x² + y2 and below the sphere x2 + y2 + z2 = z. Write a description…

A: We have, z=x2+y2            ⋯⋯1 Let us consider, x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ On substituting in…

Q: 1. The accompanying figure shows two polygonal paths in space joining the origin to the point (1, 1,…

A: Given: The function with two polygonal paths in space joining the origin to the point (1, 1, 1).…