2. Using Green's theorem, evaluate | (ev + 4x°y³ – ya) dx + (xe" + 3x*y? – 3x - :- cos y) dy, aD where aD is the boundary of the triangular region D that is defined by portions of the straight lines y = x, y = -x and y = 1, and the integration is in the anti-clockwise direction round ðD. Draw a sketch of the region. Hint: When integrating, exploit the fact that is symmetric with respect to reflection about the axis.

Q: 2. Find the surface area obtained by rotating the curve (a) y = sin(Tr) around the x-axis for 0 <x <…

A: The objective here is to find the volume of area obtained by the rotating the given curves in each…

Q: Y – 5x |dA 100 cos x + y

A: We have to evaluate the double integral ∬100cosy-5xx+ydxdy over the region D where the region D is a…

Q: Consider the double integral XY dA Vx2 + y² I where R is the region of the XY plane, given in the…

A:

Q: 7. Consider the iterated integral / T * dy dz. ly=z/2 (a) Explain why we cannot antidifferentiate e…

A: Given double integral is ∫x=0x=4∫y=x2y=2ey2dydx.

Q: For the regron x'+y si and you te temperature of the region T3 given by Texiy) = y Find te area of…

A: The temperature of the region is given by, Tx,y=xy The region is defined by below inequalities.…

Q: Calculate the double integral || x + y)?e -r dx dy where R is the square with vertices (4, 0), (0,…

A: Given: ∫∫Rx+y2ex2-y2 dx dy where R is the square with vertices 4,0, 0,4, -4,0, 0,-4

Q: Given a solid is bounded by f(x,y) = 8x² + 8y², x² +y² = 3², 3y = 4x and x = 0 in the first octant.…

A:

Q: Evaluate the integral. 3 4x A y cos z dx dy dz 0 0 0 3 4x A I ly cos z dx dy dz = U 0 0 0 (Type an…

A: We have to find

Q: 1. a) Use the cylindrical coordinates to evaluate (3 + xy)dV, where E is the solid E in the first…

A: Here is the solution of the given problem.

Q: Consider the integral dx I = 3 x+2 The approximation of I using the two-point Gaussian quadrature…

A:

Q: 1. Evaluate the triple integral //L defined by the inequalities 0<r< T, 0<y< 1,0 < z <T/6. xy sin…

A: As per bartleby instructions when multiple questions are given, only first one has to answered.

Q: . 3. Compute E y 2 z 2 dV , where E is the region bounded by x  1
y 2
z 2 and x 

A: The given curves are

Q: 3. Evaluate // (22 + y) dV where G is the solid in the first octant bounded by r = y? and r+z = 1.

A:

Q: The Fresnel integrals are used in the study of the diffraction of light and are given by: c(t) = f,…

A: Given: Her the given are c(t)=∫0tcosπ2ω2dω and s(t)=∫0tsinπ2ω2dω. By using the composite trapezoidal…

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

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

Q: Evaluate the triple integral∭E xy dV where E is the solid tetrahedon with vertices…

A: Given Data : Triple Integral ∬∫ExydV and E is the solid tetrahedron with vertices 0,0,0,…

Q: Evaluate the triple integral. 16.  ∫∫∫T  xz dV , where T is the solid tetrahedron with vertices…

A: The solid tetrahedron T with vertices 0, 0, 0, (1, 0, 1), (0, 1 ,1) and 0,0, 1. The solid region is…

Q: The dynamic equilibrium of a one-story building is described by the following equation: x(t) = xc(t)…

A:

Q: Question 12 Evalvate the to $2xy dx + (x+y) dy C is the OF y=o direction line integral by applying…

A:

Q: e dV where E is bounded by the parabolic cylinder z = 1 -y and the planes Evaluate the triple…

A:

Q: Given the curve y = (sin x)² 1. Compute the area bounded @ x = 0 to x = 2π.  a. π sq.units b. 2π…

A: Area of the bounded by the curve : If y=f(x) be an continuous function on a,b , then the Area…

Q: 3) Let F = (y+ev=,8x+ cos(y*)) Use a theorem to compute ſ F •dT where c is the boundary of the…

A: Given that  F→=y+ex,8x+cosy3 To compute ∫cF→⋅dr→ where c is the boundary of the planar region…

Q: 3. Use polar coordinates to evaluate the double integral. (a) ff sin(x² + y²) d4, where R is the…

A:

Q: Y – 5x |dA x + y 100 cos

A: We have to evaluate the double integral ∬ 100cosy-5xx+ydxdy over the region D where D is the…

Q: A- Use the separated solutions to solve the Laplace equation. in the region 0 0 given the boundary…

A: This is a question of the Laplace equation. Use the equation of Laplace to begin with, then…

Q: Consider the line integral L(- In(2² + y°\j+ tan '(2)i). dr where C is the positively oriented…

A:

Q: ¹ [ ] [ ²0 x dxdydz where E is the solid limited by Plane E z = 0,z=x+y+5 and by the cylinders x² +…

A:

Q: 3. By using Green's Theorem in the plane, evaluate | (y - sinx) dx + cos x dy where C is the…

A:

Q: 9-x2 C: Change to polar and evaluate the double integrals S (x3 + xy?)dydx V9-x2

A:

Q: (b) || 6xy dV, where E lies under the plane z 1+x + y and above the region in the xy-plane bounded…

A: A triple integral involves the integral with respect to x,y, and z. In this problem, we have to…

Q: Q2) Evaluate the triple integral. cos(:/y) dV, where G is the solid defined by the in- G. equalities…

A: Given ∫∫∫GcoszydVand π6≤y≤π2, y≤x≤π2, 0≤z≤xy

Q: y? = 1? Which of the following integrals represents the length of the upper half of the ellipse 25…

A: Topic = Application of integration  Correct answer = A

Q: 1 Evaluate the double integral| dx dy where T is the quadrilateral bounded by the lines x = 1, x =…

A:

Q: 1. Evaluate the integral |// cos (-) dV, where G is the solid defined by the G inequalities <y< , y…

A:

Q: Evaluate |(2y+cos x)dx + (5x+ sin y)dy along the positively oriented curve C. C is the boundary of…

A: We have to evaluate the integral ∫2y+cosxdx+5x+sinydy along the positively oriented curve C. Where C…

Q: 1 dA , where R lies in the second quadrant and is bounded by the graphs of y = -x, the y-axis, and…

A: Use  ,  and    to convert Cartesian form to polar form . Note : The area can represented by .

Q: Find the area of the region that is bounded by the given curve and lies in the specified sector. 4.…

A: We have to use area formula of polar coordinates with a= pi/2 and b=2pi

Q: Find the value of the closed loop integral z3 + sin z -dz (z – i)3 taken around the boundary of the…

A: GIven  ∮z3+sinz(z-i)3dz   To compute this integral we have to use the following theorem. Theorem:…

Q: Let R be the region inside the ellipse x = 4 cos 0, y = 3 sin and outside the circle x = 2 cos 0, y…

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: c) Evaluate the integral e* (x + 1) - dx, VI – x² using 3-points Gaussian quadrature.

A:

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

A:

Q: 7. Evaluate the integral I | sin(9.r² + 4y²)dA by hand, using an appropriate change of variables,…

A: The given question can be solved as shown in step2

Q: Evaluate the triple integral -3pi D x dV where E is the solid bounded by the paraboloid x = 6y² +…

A: Solution is given below:

Q: I// sin(x²+y²) dV , where E is the region E = {(x, y, z) : x²+y² < 1, 0 <z< 1}. E

A:

Q: Calculate the iterated integral. 1/2 (y + y2 cos(x)) dx dy -3Jo

A:

Q: The electric potential at each point (x,y) in the region R: x 2 0, y s1 is given by v - 48xy – 32x -…

A:

Q: (1) Sketch the regions of integration. (a) 7/2 Jo r3m/2 (b) (1r, 0)r drdo 3n/4

A: Consider the equation

Q: 3. Use Greens theorem to evaluate (e* + y²)dx + (e' + x²) dy where c is the boundary of the region…

A:

Q: 1. Evaluate the line integral I = = [v¢4x° + y*) dx + x(2x° + 3y*) dy] around the ellipse x/a? +…

A: The given line integral is, I=∮y4x2+y2dx+x2x2+3y2dy We know the Green's theorem:…