Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting (1, 1, 1) to (1, 2, 4).

Q: By using Green's theorem evaluate £ (Ary – y") dr + (** + 3ry) dy where C is a closed curve starting…

A:

Q: find the tangent plane to the surface z = Vy-x at the point (1,2,1)

A: Equation of tangent plane is z-c = df/dx(x-a)+df/dy(y-b) at point (a,b,c)

Q: 1. Use the Stoke's Theorem to evaluate y'dx + xy dy + xz dz, C where C is the bounding curve of the…

A:

Q: Let C be a circle lying in the plane x+ z = 2 with center at (1,0, 1). Evaluate the line integr V2…

A:

Q: Use Green's theorem to evaluate , (5xy + x2 + y?) dx + (x²-y)dy where C is a closed curve that is…

A:

Q: Find the line integral ye-xz ds Here, C is the path r(t) = [t,3t,-6t], 0sts In(8) ·

A:

Q: Suppose the closed curve C in R? starts at the point (-1,1), goes along the parabola y = x2 to the…

A:

Q: Evaluate the line integral . xy ds, where Cis the top half of the circlex +y = 16.

A: The given integral is ∫C xy2 ds, where C is the top half of the circle x2+y2=16. To find: the value…

Q: Let P be the plane that is tangent to the equation xyz + 260 = z' + 30x + 26y at the point (7,7,4)…

A: The tangent plane to the surface is a plane that only touch the surface at the point.

Q: Compute the line integral ∫_C F · dr, where F (x, y) = 〈−x^3, xy〉 and C is part of circle x^2 + y^2…

A:

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy'dy, where C is the triangle with…

A:

Q: Evaluate the line integral [x(1 с where C is given by the line r(t) = (1 + 3t, 2t), 0≤ t ≤ 1. x(1 +…

A:

Q: Find the positively oriented simple closed curve C for which the value of the line integral C(y^3…

A:

Q: Evaluate the line integral 32xy ds, where C is the portion of the parabola y = x² from (-1,1) to…

A:

Q: Compute the line integral -xdy + ydx 2(x-y)2 +4xy where L is the circle (1-x)2 + y2 = 3 with a…

A: We can write the given line integral as ∫L y2x-y2+4xydx+-x2x-y2+4xydy, where L is the circle…

Q: Evaluate the line integral f. (2x – 4xy)dx + 3xy – 2x³y)dy over the straight path L joining point…

A:

Q: Consider points P and Q in the x-y plane, with P = (1,0) and Q = (0,1). The line integral %3D Q…

A: The line joining P1,0 and Q0,1 is     x-11-0=y-00-1=t(say)⇒x=t+1,y=-t⇒dx=dt,dy=-dt

Q: Evaluate the line integral / x°z ds, where C is the line segment from (0, 2,4) to

A: Line integral: - The function to be integrated is determined along a curve in the coordinate system…

Q: A point moves along the intersection of the elliptic paraboloid z = x + 3y² and the plane y = 3. At…

A: z=x2+3y2 and y=3 Point of Intersection : Curve z:x2+3y2=x2+3(3)2=x2+27 dzdx=2x+0 =2x

Q: Calculate the line integral F(x, y, z)=2xy^2 i+(2yx^2 +2y)j     on a path from (0, 0, 0) to(3, 1, 1)

A: We need to compute the line integral: F(x,y,z)=2xy2i+(2yx2+2y)j on a path from the point: 0, 0, 0 to…

Q: Evaluate the line integral xyld x + (x²+y)dy] +. where c is the saruare farmed by the lines

A: Let us consider a line integral ∫CP dx+Qdy then it can be written as  ∫D∫∂Q∂x-∂P∂y dA  Where, P and…

Q: Evaluate the line integral ∫Cydx+xdy along the curve y=x2 from the point O(0,0) to the point A(1,1)…

A:

Q: Let f(x, y) = 2x and C be the line segment on the plane from (0, 0) to (3, 4). Evaluate the line…

A:

Q: Find the slope of the tangent line at (-1, 1,5) to the curve of intersection of the surface z = x²…

A: According to question, we have, z = x2 + 4y2        .......(1)

Q: Evaluate  ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9  with positive…

A:

Q: 4. Evaluate the line integral / r y* ds where C is the right half of the circle r² + y² = 16.

A:

Q: Evaluate the integral [(3x – 2y + z)ds where C is the curve, C, and C, as shown. 3 y²+ z² 2 G 2* 3…

A: Given that ∫c3x-2y+zds. The objective is to find the integral value. ∫c3x-2y+zds=∫c13x-2y+z…

Q: • Use Green theorem to evaluate the Line integral $(x-3y) da +(42+y) dy whe re C, the Yectungle with…

A:

Q: Sketch the plane curve. r(t) = 4ti - tj, [0, 4] y y 4 4 2 10 15 10 15 o-4 o-4 y y 4 X 5 10 15 10 15…

A:

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy*dy, where C is the triangle with…

A: The given triangle C with the vertices (0,0),(1,0) and (1,2) is as as shown below.

Q: Find the equation of the tangent plane for the surface z = xy at the point (-1,2,-2) O z=x-2y+2…

A: The equation of tangent plane for the surface z =f(x,y ) at point (m,n,f(m,n)) is given as:

Q: Which type of surface does the curvex= cos(3t),y= sin(3t),z=t, lie on?

A:

Q: Exercise (2): Calculate fe xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to…

A: we have to following integral

Q: Use the Green's Theorem to evaluate the line integral 2xydx + x² – y² dy where C is the curve…

A:

Q: Evaluate the line integral |(3="y + (1- y):) dz + (z² – z2) dy + z(1 – y) dz where C is any curve…

A:

Q: Find a vector normal to the surface x2 + yz = 5 at (2, 1, 1).

A:

Q: Evaluate the line integral T, (“+y) dx + 4 ry dy where L is the path, -> r) = (3 t, 2) with 0 <t <1.…

A: We evaluate given integral

Q: Evaluate F-di where F(x, y, z)=1li +(x+yz)j+(xy- Vz 3x+2y+z=6 in the first octant. C is a positively…

A: Here we have to evaluate ∫CF→·dr→=? Where F→(x,y,z)=i^+(x+yz)j^+(xy-z)k^ and C is the boundary of…

Q: : The curve C is intersecting by 12r² +9y? = 36 cylinder and - V3r plane from the point (0, 2,0) to…

A:

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: Compute e3z dz , Jy (z – 1)2(z+ i) where +y is any simple closed curve oriented counterclockwise…

A:

Q: Evaluate the line integral Sa y dx +z dy, where C is the line segment from (4) (-1, 1, 2) to (2,3,…

A:

Q: Find the tangent plane to the elliptic paraboloid z = x² – 2y at the point (1,1,1). - O A. z = 2x –…

A: Given that: z=x2-2y2      and point is:   (1, 1, 1)

Q: Evaluate the line integral x²z ds, where C is the line segment from (0,6, 2) to (8, 5, 1).

A:

Q: Let C be a curve given by the intersection of the surfaces z = x2 +y2;z = 3−2x . The value of the…

A:

Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

A:

Q: Evaluate the line integral y dx + x dy where C is the parameterized path x = = ² , y = t° , 3 -ydx +…

A:

Q: Find the equation of the tangent plane Pr to the surface S:r +y +z2 = 9 at (2,-1, 2) | O PT:4x + 2y…

A: The equation of the tangent plane to the surface f(x, y, z)=0 at x0, y0, z0 is defined as: fxx0, y0,…

Q: Find the equation of the tangent plane to the surface xy + 2yz + 3zx = 11 %3D at the point P (1, 1,…

A: