Calculus II TU Board null question paper

• a) Evaluate the integral by reversing the order of integration: mp (si Sy Je ( = ) dydx 1 i-x xty
• b) Evaluate the integral: J J i} e* dzdydx 00
• c) Find the volume of the solid whose base is the region in the xy-plane that is bounded by the parabola y = 4—x? and the line y = 3x, while the top of the solid is bounded by the plane z = x + 4.

• a) Solve by using power series method: y" — 4xy’ + (4x? — 2)y = 0.
• b) Solve the Bessel’s equation x*y" + xy’ + (x?—n)y =0.

OR

• i) Express x’ — 4x + 2 as a Legendre polynomial by using Rodrigues formula.
• ii) Sketch the graph of Po(x), Pi(x), P2(x) and P3(x) in a single graph.

• a) Solve the initial value problem by using Laplace transform: y" + 2y' + 2y = Ssint, y(0) = y'(0) =0
• b) i. Find the Laplace transform of: cosh at sinat . oe . . s :
• ii) Find the inverse Laplace transform of @apetre Using the convolution theorem.

• a) Ifp=x3+y3 +23 - 3xyz, find: div(gradp)
• b) Find the directional derivative of ¢ = x? — y’ + 22? at the point P(1, 2, 3) in the direction of line PQ, where Q is the point (5, 0, 4).
• c) Find the work done by the force F=(2y+3)i +22] +(yz—-x)k whena paiticle moves from (0, 0, 0) to the point (2, 1, 1) along the curve x=2t, y=tz=¢, = -

) Find the flux of F =zi+xj+yk over the portion of the surface x?+y?= 1 in the first octant between z=0 to z= 2.

• b) State Stoke’s theorem. Use it to evaluate GF. d7, where F=(z,x,y) and c is the surface of upper half of sphere x? +y? +z? =a? in xy-plane.

OR

State Gauss divergence theorem. Evaluate ff, F.7ids , where F = (4x,x?y,—-x?z) and S$ is the surface of the tetrahedron with vertices (0,0,0),(1,0,0), (0,1,0), and (0,0,1), by using the Gauss divergence theorem.

• a) Find the Fourier series of the function f(x)= |x| for -2 <x <2. 7.
• b) Expand the function f(x) =x? in the interval O<x<z in half-range Fourier cosine series and deduce that nm 4,4,3 Faitsgtgtet

Solve: (Any two) 2x5

• a) Find the general solution of u, + uy — u = 0.
• b) Derive one dimensional traffic flow model using convection law.
• c) Show that the integral an ery 42? (dx — dy + 2zdz) is exact and evaluate the exact value. a