Use Green's theorem to calculate §. -y² dx + xy dy where C is the vertex triangle (1,3), (4,0) and (4,3). 3. X 2 3 4

Q: r(t) = ti + (2tt2) intersect the paraboloid (2) At what points does the curve r(t) z = x² + y²?

A: We are asked to find the intersection point.

Q: + y? )dx + (x² – y²)dy , where C is the triangle with vertices (0, 0), (2, 1), and (0, 1), sitive…

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Q: Q.2)(A) Find a formula for the area of the triangle in the xy-plane with vertices at A(0, 0), B(a1,…

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Q: Evaluate ʃC F dr where C is the parabola y = x2 from (0, 0) to (2, 4) and F (x, y) =< −ysinx,…

A: Given a curve C is a parabola from (0, 0) to (2, 4)

Q: Part (a) valuate 36e dz, where C'is the polygonal path onsisting of the line segments from z = Oto =…

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Q: Let C C R? be the path from (3, –2) to (3, 2) along the graph of the curve x – 5x?y? – 7x2 = 0.…

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Q: Let R be the region in the fourth quadrant bounded below by the parabola y=x²-3 and above by the…

A: Use double integral

Q: Verify Green's theorem for F =, where C is the part of the parabola y = x², and the line y = 4 for…

A: If be a positively oriented,  closed curve  enclosing the region D Then for a vector field F→=P,Q…

Q: The parametric form of 3x - 2y + 10 = 0 is: O x = 2t y = 5 + 3t O x = t y = 5 + t O x = 3t y = 5 +…

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Q: What is dyldx at the point (1,2) of xy^2 + 2xy 8 ?

A: The given equation is: xy2+2xy=8...(1)

Q: Find the linear approximation at x = 0 to Write %3D 2- your answer in the form y = Ax + B. %3D

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Q: Evaluate SF dR given that F (x, y) = 2yî + (xy – 2x) %3D and C is the curve in the third quadrant…

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Q: D A Find thepolar eavation for the cartesian curve (x²ty) - A)r-4 cos o =4x. Br°=4 cos e c)r?=4cos o…

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Q: .2 y = z, -2 < z < 2 %3D (x - 2)² + 4y2 = 4

A: (b)  The given equation is y=z2, -2≤z≤2. Observe that y-coordinate will always be positive and…

Q: Find a point on the curve y = x² that is closest to the point (18, 0). %3D AY y =2 (x, y) (18, 0)

A: Find a point on the curve y=x2 that is closest to the point  Here use distance formula…

Q: 4. Let C be the unit circle in the xy-plane, and S be the boundary of the square with side 1,…

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Q: Evalwate f, the inaicated point. (Y.y) and Fy y,y) CY.y) at 3. at (l,3)

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Q: Sketch the graph of y = m(r) in Figure 2. %3D 음-3, rS-1 m(x) = 4 %3D 1<r<5

A: Given,          mx=13x-3x≤-141&lt;x&lt;5

Q: The Normal to the curve 2x2 – y? = 1 at the point (1,1) is: %3D 3 y =-*+ y =x+ y = 2x + 1 %3D P. y =…

A: To find The equation of normal to the curve 2x^2-y^2=1 at the point (1,1)

Q: dx, using two and three points Gauss Quadrature Q4 (a) Approximate the value of

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Q: 8. Calculate fo F • dr, where F(x, y) = y² i + (6x + 6y)j, and C is the portion of the parabola 2y²…

A: Given curve is 2y2-x=1 and points are (1,-1) to (7,2)

Q: 12. Evaluate 3x ds where C is the parabolic are along y from P(0,0) to Q(2, 2).

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Q: 3) For the parametric system, find dy d2y When is the graph concave up or down? and dx dx2 x = t2 -…

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Q: (A) Evaluate" [(x² + y)dx + (x+1+d along the path ABCA shown in figure C(0,3) A(0,1) x=3sint y=3cost…

A: What is Line Integral: When we integrate a function along a straight line or a curve inspite of…

Q: 3. Find an equation of the tangent plane to the paraboloid defined by z= 5x +4y? at the point…

A: The given equation is; z=5x2+4y2 at the point (-1,1,9).

Q: I need a 3d sketch of the equation  Z=x+xsiny+xy

A: z=x+xsiny+xy

Q: Part (a) Evaluate 4e*dz, where C is the polygonal path consisting of the line segments from z = 0 to…

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Q: Vf(x, y) = 4y sin(xy)i + 4x sin(xy)j, F = V f(x,y), and C' is the segment of the parabola y = 5x²…

A: According to the fundamental theorem of line integral, for a smooth curve C given by r→t, a≤t≤b,…

Q: (b) Find the equation of the tangent plane to the hyperboloid r2 + 3y? – 322 = 1 at the point (1,…

A: We have to find the equation of the tangent plane to the hyperboloid x² + 3y² − 3z² = 1 at the point…

Q: 10. If ZYXW is an isosceles trapezoid with WY = 15x-2 and XZ = 9x + 10, find x. X 1 partn Z Y

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Q: Iculate | y?dx – yz²dy – sen(z)dz, where Cis the path connecting the point A(-2, 2, 4)to the point…

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Q: Find the point on the line y = x closest to the circle of radius 1 and center (5, 2) using Lagrange…

A: Please refer to the image below

Q: The length of the curve y = Ey3/2 between x = 0 and X 1 is %3D

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Q: Find the coordinates of the point(s) on the curve 2y = 10 – a² that are closest to the origin. а) О…

A: Given that

Q: 4 du + 3, du -o is dassified as Jhe P.O E dx² © Hyperbotic Elliptic Parabolic there of

A: 4∂2u∂x2+3∂2u∂x∂y=0

Q: 17. Evaluate F · dš if F(x, y) = (y*, 2xy*) and & is the ellipse x? + 2y? = 2 (with counterclockwise…

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Q: Part (a) Evaluate 100e dz, where C is the polygonal path consisting of the line segments from z =…

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Q: Use Green's Theorem to evaluate -1 (y? tan a?) dx + (3.xy + 4x – In(y² + 4)) dy, where C is the…

A: The green's theorem states that "line integral over a closed curve C…

Q: 4x2 dx - xy dy 16x2, y = o from (-1, 0, 16) to (1, 0, 16) (d) dz, where c is the parabola z

A: To evaluate the line integral along the arc C of the given parabola

Q: Find the point on the line y = x closest to the circle of radius 1 and center (5, 2) using Lagrange…

A: First I have find the function f(x,y) and then decide the constraint.Then using Lagrange multipliers…

Q: Let Q = 4x2 + 6xy+ 4y? . The points on the curve Q = 7 which are closest to the origin are: %3D %3D…

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Q: Verify Green's theorem for F =, where C is the part of the parabola y = x², and the line y = 4 for…

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Q: Let f(x, y) = x? + 4y. Find the extreme values of f on the ellipse x? + 2y? = 1.

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Q: Evaluate | 9+x* dx+xy*dy along the positively oriented curve C where the C is the boundary of the…

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Q: Evaluate I=∫C(sinx+4y)dx+(5x+y)dy for the nonclosed path ABCD. A=(0,0), B=(1,1) ,C=(1,2), D=(0,3)

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Q: 6. trapezoid DEFG with vertices D(0,-3), E(1, 3), F(3, 3), and G(4,-3) in the y-axis ty O X D' =_ E'…

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Q: find the location of a point xyz that satisfies the condition I) xy<0, z=3 ii) |x|<2

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Q: c) By using Green's theorem evaluate $ (4xy – y°) dx + (x³ + 3xy) dy where C is a closed curve…

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