1.108. Find F•dr where F = (z - 3y)i + (y - 2x)j and C is the closed curve in the zy plane, z = 2 cos t, y = 3 sin t, z = ɔ from t=0 to t= 27. Ans. 67

Q: 3) Consider the curve C in R³ given by: r(t) = (t + In t, t – In t, 4t'/²) con 1<t< 2 An expression…

A: Formula: If r(t)=x(t), y(t), z(t), a≤t≤b is parametrization of a curve C then the line integral…

Q: 44. Let r(1) = (r2, 41 – 3). Find T(t) and N(1), and show that the de- composition of a(1) into…

A:

Q: (d) Let P = xy², Q = x²y and C denote one petal of the four leaved rose defined by r = -450<.…

A:

Q: Let f (z)=Z. Find Jf (z )dz from z = 0 to z =4+2i alongthe curve C given z 2i to z =2i.

A: We have to solve the integral ∫Cf(z)dz The given function f(z)=z¯. The curve is given from z=0 to…

Q: (3) Compute f ds over the curve specified. (a) f(x, y, z) = 2²,r(t) = for 0 for 0 < t < 1

A: Consider the equation

Q: dr. Let f(z) =z. Find from z-0 to Z-4+2í alongthe curve C given Z=21 to z=2i

A:

Q: Q3:Let w = In(x² + y2 +z2) %3D x = ue" sin u y = ue' cos u %3D z = ue" Find : aw A) by chain rule du…

A:

Q: 1. Find the work done by F(x, y, z) = (-y,–x, z) on a particle that moves along the curve R(t) (2…

A: As per our guideline we are supposed to answer only one question you can repost other questions in…

Q: 13. Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A:

Q: Let C be a portion of a curve from (2,0) to (2,2). Let F(r,y) = (tan(x²) + e²v¸ 2re²v

A: Introduction: A vector field is conservative if ∂P∂y=∂Q∂x, Where f(x,y)=Pi+Qj.

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

A:

Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A:

Q: 7. Given the space curve er(t) = (4 sin t ,4 cos t , 10t), Find T, N, B, and t (torsion)

A:

Q: 9. let C G TR? be Simple Close d curve bounding the positively orierned a reyi'on D. IF Area cO)=|…

A: Given that, Area(D)=∫Cx2+2xy+2y+eydx+∫CAx+x2+xeydy

Q: [39] Consider the heat conduction in a circular cylinder: x² + y² < a²,0 < z <b. Assume that the…

A: Given Data: Heat conduction in a circular cylinder  x2+y2&lt;a2, 0 &lt;z&lt;b To Find: (a) Boundary…

Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: 11. Evaluate the line integral F dr where F (x,y) = xy'i+vi and the curve C is the shortest path…

A:

Q: Which of the following expressions are true? 1. 2z + 4y – z = 5 is the equation of a tangent plane…

A: The equation of a tangent to a function is given by the formula . Here,  is the point at which the…

Q: 6. Find F' and dF[a; h] where: In ¤1 (a) F(x1,x2) and hence show that if v is tangent to the curve…

A: (a.) Given: Fx1,x2=lnx1x2e2x1x22cosx1 To find: F' and dFa;h To show: If v is tangent to the curve…

Q: 2. Let L be the line tangent to the space curve represented by r(t) = (t² – In(t))i+(e²-t – t)j+…

A:

Q: 5. True or False: suppose that for any simple, closed curve C, ſF · dr = 0 where F(x, y) = (f(x, y),…

A:

Q: 8. (a) Show that F(x, y) = (ye + sin y)i+ (e" + x cos y + 2y)j is conservative and find a function f…

A:

Q: Consider curve C at R parameterized by: r(t) = cos(2t) î + tf+2 sin(2t)k, with 0 stsT An expression…

A:

Q: (6) Find the integral of the function piecewise linear path from (1,1) to (2,2) to (0,2). b) 22.3…

A: Given the function Fx,y=ey-x,e2x

Q: Consider the curve C in RS parameterized by r(t) = sin(2t) î + cos(t) ĵ+t k, con 0 <t < 2n An…

A:

Q: 3. Find the work done by F over the curve in the direction of increasing t. = (2,x, y) = (sin t, cos…

A: Write the vector field F=zi+xj+yk The equation of the curve. rt=sinti+costj+tk Differentiate…

Q: 4. Evaluate the line integral f x siny dx + y dy, where C is the arc of r(t) = ti + t²j from point…

A: Line integral

Q: 1. Sketch in the xy - plane the domain of each function : (a) ƒ (x, y) = (Vx – 1+ Vy +1) · In (x² +…

A:

Q: 11. Compute the length of the curve r(t) = (2t, In(t), t²) over the interval 1 <t < 4. (cos(t),…

A:

Q: If ƒ` = (x+1)√/x² + y i + ² tan^¹(3x−y)j and ğ=- y at (0,1,²7) 4 find: fxcurl ğ +(g.V)ƒ at x + y…

A:

Q: 1. Evaluate the line integral of E = xax - y ay along the segment P₁ to P2 of the circular path as…

A:

Q: 13. Let F (x, y) = (1(²y²)x, af(x²y? , a(y²) Y) and suppose that C is a smooth curve parametrized…

A:

Q: (7) A particle moves along the curve 7= (13 – 4t)i + (t? + 4t)j+ (8t² – 3t®)k, where t is the time.…

A: Tangential acceleration is defined as the rate of change of velocity vector where velocity vector is…

Q: 3.5 Given that H = x'a, + ya,, evaluate H· dl, where L is along the curve y = x from %3D (0, 0) to…

A:

Q: Let f (z)=z. Find zpl 2) from z =0 to z =4+2/ alongthe curve C given z 21 to z =2i.

A:

Q: 1. Sketch the direction field and solution curve for the given DE and condition. Sketch the…

A:

Q: (7) Let w(x, y, z) = xy cos z, where x = t,y = t², and arcsin(t). %3D dw Find dt

A:

Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

A:

Q: 11 Calculate the line integral f Gy)ds for the given junc rion and path xiyl=xy KHt)=<3+,2t=-37 for

A: r(t)=&lt;3t,2t-3&gt;=&lt;0,-3&gt;+t&lt;3,2&gt; x(t)=3t  and y(t)=2t-3 ds=32+22dt=13dt 0≤t≤1

Q: Two waves on a string are given by the following functions: yı (x, 1) = 4 cos(20 – 30x) (cm) y2(x,…

A:

Q: (1) Find the area of the curved surface of the upper half of the sphere x2 + y² + z² = a²(a > 0) on…

A:

Q: 5. Let u(x) be a differentiable function with u(1) = 2 and u'(1) = 3. Find an %3D equation of the…

A:

Q: 1. Consider the curve C = C₁ UC₂, where C₁ is the line segment from (0, 1.1) to (1,1,0) and C₂ is…

A: Given Data: The given curve is C=C1∪C2, where C1 is the line segment from 0,1,1 to 1,1,0 and C2 is…

Q: 1. Given A = (yz + 2r)i + xzj + (xy + 2z)k. Evaluate , A• dr along the curve x +y = 1, z= 1 in the…

A:

Q: 23. Give a formula that gives the length of the outer loop for the curve: r= 4-8 cos(0) (Hint: Graph…

A: r=4-8cos(θ)For loops crossedr=04-8cos(θ)=08cos(θ)=4cos(θ)=12θ=π3,5π3(Have symmetry from 0 to π and π…

Q: 28. Suppose you follow the spiral path C: x = cos t, y= sin t, and z= t, for t20, through the domain…

A: NOTE: You have posted a question with multiple subpart, we have given answers to the first three…

Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: (b) Given  = (x²y)î + (xz – z²)f – (xyz)k, determine the following at point (1, 1, 2): (i) V- Â.…

A:

Q: The area under the curve y = 3z bounded by the linesI = 3 and z= 7 above the z- azis is O 174.00…

A:

Q: Question 4 Evaluate the line integral: (i) of T(x) = 4x^3 along the line segment from (-2,1) to…

A:

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.