1.13 Evaluate the line integral $ x*dx + xydx where C is the triangle with vertices at (0, 0), (0, 6) and (6,0) vith positive orientation.

Q: Verify Green's Theorem by evaluating both integrals Lv² dx + x² dy dA ax ay for the given path. C:…

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Q: 1.10.6 Show, by expansion of the surface integral, that S, do x V lim = V x V. sdt-0 fdr Hint.…

A: To show that lim∫dτ →0∫sdσ×V∫dτ=∇×V

Q: 1. Calculate the integral 1 y-stnx)dx + cosy dy a) With directly (classical), b) With the Green's…

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

Q: 2. Evaluate each of the following line integrals in two ways∗ . (a) Z C1 F~ 1 · d~r, where F~ 1(x,…

A: Given vector field is F→(x,y)=(2x-cosy)i^+(xsiny)j^F→.dr→=(2x-cosy)dx+(xsiny)dy C1 is the…

Q: Evaluate the line integral Ĥ. dř where Ĥ = e" î + [xe" + In(2)] ĵ + º Â and the curve C is defined…

A: Given problem:-

Q: 2) Evaluate the following line integral: . 0+z)dx + (x + z)dy + (x + y)dz where C consists of the…

A: C1: (0,0,0) to (1, 0, 1)x-01-0=y-00-0=z-01-0=tx=t, y=0, z=tdx=dt, dy=0, dz=dtI1=∫01[(0+t)+(t+0)]…

Q: 5. Evaluate the integral xdx+ (x+y)dy where C is the curve x 1+y from (2,1) to (2,-1).

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Q: 3 2(x²+1)7 Write down an integral that gives the ARCLENGTH of y on the interval from x = 0 to x = 2…

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Q: х cos 4x dx; и 3D х, dv cos 4x dx =

A: Given that  u = x ;  dv = cos 4x dx and we have to integrate by parts the given integral that is ∫x…

Q: 7. Evaluate the line integral by using Green's Theorem feve'dx +2e*dy C is the rectangle with…

A: Green's Theorem, ∫CPdx+Qdy=∬D∂Q∂x-∂P∂ydA Compare ∫CPdx+Qdy=∫Cyexdx+2exdy Implies that, P=yex, Q =…

Q: 2. Evaluate each of the following line integrals in two ways*. F (x, y) = (2x – cos y) i + (x sin y)…

A: 2. (a) Directly: The equation of the curve C1 is x-4+y5=1( intercept form), or, y=54x+5. The…

Q: Q3)Verify the Greens theorem to the integral f(2x*y-x)dy +(2yx- y)dx where C is atriangle its…

A: Green's theorem

Q: 1. Give iterated integrals for r (x² + yz)dV where E is in the first octant bounded by the planes: y…

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Q: 1. Evaluate the line integral of E-xa, -y a, along the segment P; to P; of the circular path as…

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Q: 1. Evaluate the given line integral, I (-y,x – y) · dR, where C is the upper part of 4.x? + 9y? = 36…

A: we have to find the line integral  ∫C-y,x-y.dR→ where c is part of the ellipse 4x2+9y2=36 from…

Q: 5. Evaluate the line integral (r+2y) dr+x² dy where C consists of line segments from (0,0) to (2, 1)…

A: In order to solve this question by the formula of line integral .

Q: 9. Use Green's Theorem to evaluate (e* +y²) dx +(e" +x²) dy where C is the boundary of the region…

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Q: 22 Calculate the value of the integral over the following C curves. a) C:l2+1| = 1 1, %3D c) C:l오+1…

A: we have ∮Cz2z2-1 a C: z+1=1⇒C:x+iy+1=1⇒C:x+1+iy=1⇒centre is -1,0 and radius is 1 Thus, only pole…

Q: Verify Green's Theorem by evaluating both integrals ax dA ду for the given path. C: rectangle with…

A: From: (0, 0) to (5, 0)y=0, dy=0∫ y2 dx+ x2 dy =0From: (5,0) to (5, 9)x=5, dx=0∫09 x2 dy= ∫0925…

Q: (b) Evaluate each of the following line integrals: (1) xds, where C is the portion of the curve x…

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Q: 2. Set up an integral to find the exact length of the curve defined by 1 f(r) 1 In r %3D where 1<r <…

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Q: 1. Evaluate the given line integral, |(-y,x – y) · dR, where C is the upper part of x² + y? = 1 from…

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Q: Verify Green's Theorem by evaluating both integrals Lr dx + x* dy = / dA ду x² dy ax for the given…

A: In this question we verify Green's Theorem                 i.e, verify integral  ∫C y2 dx +x2 dy =…

Q: 19.23. Evaluate the integral , dz/(z – 1 – i) where y is: (a) the line joining 3; and (b) the path…

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Q: Use the Fundamental Theorem of Line Integrals to calculate d exactly. 7-7+4y²f arqund the top of the…

A: Given

Q: 1) Evaluate the triple integral. | 6xy dV where E lies under the plane z = 1+x + y and above the…

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Q: 4. the value of the given integrals | z - i | = 2,(z € C) around the circle compute for positive…

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Q: 11 Calculatk the line integral , f (x,y)ds for the given junc rion and path xiyl=xy Ht)= for 0st<4…

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Q: 5. Evaluate the iterated integrals. a. b. [²³² (x+y)dx dy S S S xyz dx dy dz

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Q: 2. Evaluate. the fallowing.integrals.a.fter.seveoiag.the areler.of.integratiam. dydx Ans..In)/4 C.…

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Q: Evaluate the double integral y sin(2³)dA where R is the triangular region with the three vertices…

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Q: 8. Evaluate the line integral x dx + xedy where C is the closed path along the triangle in the…

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Q: Verify Green's Theorem by evaluating both integrals √< y² dx + x² dy = [] (ON - 3M) de dA ?х ду for…

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Q: 9. Evaluate the integral = R from z₁ = 1 to Z2 = -1. where C is the above half of the circle

A: Given is the integral ∫Cdzz¯. where C upper semicircle. |z|=R, z1=1 to z2=-1

Q: 19 9% Set up the iterated double integral S S dy dæ as dxdy such that: 0 0 i. the limits of x ii.…

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Q: Evaluate the line integralſ (x + 2y) dx + x² dy, where C consists of line segments from (0,0) to (2,…

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Q: 3 For questions 8 and 3. consider the line integral Sx²ydx+xydy 1439) Evaluate the integral, if C rs…

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Q: 3. Consider the line segment L = [-1-i, 1 + i]. Calculate the value of the integral 1₁² z² dz.…

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Q: 11. Use Green's Thm to evaluate the line integral if the curve C consists of 3 line segments, from…

A: Solution:-

Q: Verify Green's Theorem by evaluating both integrals L v? y2 dx + x2 dy = dA ду ax for the given…

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Q: Verify Green's Theorem by evaluating both integrals v² dx + x² dy dA ax ду for the given path. C:…

A: This question is related to vector calculus solved in step 2.

Q: 15. Use the indicated change of variables to evaluate the double integral ||18xy dA where R is the…

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Q: 15. Evaluate the following line integral if C is a piece of the cubic function y=x^3 from (0,0) to…

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Q: 7. Verify Green's Theorem for O (6+ x²) dx + (1 – 2xy) dy where C' is shown below by (a) computing…

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Q: O Evaluate the line integral over The Cur ve C. SCx+y+2)dx +6x +y-2)dy+ (x-y+2)dz where c is the…

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Q: 3.4.6 Evaluate the line integral I = =f( (xy dx + x dy) C where C is the closed curve consisting of…

A: We have to solve the given problem

Q: 4. Evaluate the double integral D where D is enclosed by the curves y = a², y = 3x.

A: First we will draw the graph for given curves from that we will find the limits of x,y as shown…

Q: 17. 6xy dV, where E lies under the plane z = 1 + x + y and above the region in the xy-plane bounded…

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Q: 2. We have seen that the FTC for line integrals implies fc Vf - dĩ = 0 for any smooth closed path C…

A: FTC for line integrals implies ∫C∇f·dr→=0 for any smooth closed path C and any smooth function f (a)…