BSc Mathematics SEM V 2016 17 2016-17 Maths Integral Calculas Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 From Question 1,2 and 3, Attempt any one from part(a) and any two from part(b)
2. Q3 From Question 4, Attempt any THREE
3. Q4 Figures to the right indicate marks for the respective parts 1 a Define triple integral of a bounded function f: R where Q = [a2, b2] x [a3, b3] is a rectangular box in Further show with usual ii State and prove Fubini’s theorem for a rectangular domain in R? b State the change of variables formula for triple integral, stating clearly the condition under which it is valid. Use it to express the triple integral — dzdydx in spherical co-ordinates Evaluate J by reversing the order of integration. Sketch the region of integration fff, y where S is the solid enclosed by the planes z = 0, z = y and the parabolic cylinder y = 1 — x? 'V ff, dA where R is the region bounded by the lines y = 2 a Let f differentiable scalar field defined on U 8 in R”. Suppose C be a closed curve in U, with parameterization r(t),t [a,b]. then prove that $. =0 ii State and prove Green’s Theorem for a rectangle. Further find the area of the region between two concentric circles of radii where b 1. F a continuously differentiable function defined on a simply 12 connected region D in R*. Show that $, Pdx + Qdy = 0 around every closed curve C in D if and only if = V (x,y) li Byaluate f where F (x,y,z) = and C is the circle of radius 1, with centre at the origin and lying in the yz, plane, traversed counterclockwise as viewed from the positive x axis Ml Show that the line integral (y? + 2xy)dx + (x? + 2xy) dy is path independent. Further evaluate the line integral iv Use Green’s theorem to find the area of the region
4. Q3 State and prove Stoke’s Theorem for an oriented smooth, simple parameterized surface in R? bounded by a simple, closed curve traversed counter clockwise assuming general form of Green’s Theorem 8 marks
   • Q.P.Code:05711 ii For the surface 7(u, v) described by the vector equation 7(u, v) = differentiable on T, define the fundamental vector product x If Cis a smooth curve lying on the surface, C = (t)), [a,b] > T, then show that a normal to C at each point Further assume S and C satisfies the hypotheses of Stokes’ Theorem and f,g have continuous second order partial derivatives. Prove with usual b Assuming S and V satisfy the conditions of the Divergence Theorem and 12 scalar fields f and g, components of F have continuous partial derivatives, fA is unit outward normal. Prove
   • p) |V| = = fidS where F = xt + yf + zk and |V| = volume of V i Using Stoke’s theorem evaluate ff curl F.AdS where F(x,y,z) = xf+ z?f+y?k and S is the plane surface x + y +z lying in the Evaluate ff, yds , where S is cylinder Gauss Divergence Theorem to evaluate where F(x, y,z) = (y—x,z—y,y — x) and S is the cube bounded by the planes 4 Spherical coordinates find the volume of the solid region bounded 15 by the surface p = Find the area of the region R bounded by the curves y = sinx and y = A particle starts at the point (-2, 0) moves along the X-axis to (2, 0) and then along the semi circle y = V4 — x? to the starting point. use Green’s theorem to find the work done on this particle by the force field F(x, y) = x 1V Evaluate the line integral of f(x, y,z) = along the path parametrised y Find the surface area of S which is parametrically given by Vl Using Stoke’s theorem evaluate F.dr where = (x? + Cis the boundary of the rectangular lamina in the xy plane. Bounded by the lines x = 0,x = 2,y = 2 and y = 5 oriented counter clockwise as viewed from above

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