BSc Mathematics SEM V ATKT MATHEMATICS INTEGRAL CALCULUS 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
4. Q1 aii Prove that a continuous function is integrable on a rectangular domain. ii State the Change of variable formula for a triple integral stating the conditions under which it is valid. Explain further , how will you use it to a express triple integral in spherical co-ordinates ( p, b i Fubini’s Theorem to evaluate {f s f where f(x,y) =x +y and S is defined 12 it Use polar co-ordinates to find area of a region S in the first quadrant of circle x? + y* —8y = 0 below the line y = V3x iit Using cylindrical co-ordinates, find the volume of solid S bounded above by the paraboloid z = 5 — x? — y? and below by the paraboloid z = 4x? + 4y? iv State Leibnitz rule for differentiation under integral sign. Hence find g’(x) for g(x) = log(x? + dy. Verify your answer by finding g(x) by direct 8 marks
5. Q2 ai i Let f be a continuously differentiable scalar field defined on an open set U in IR”. Suppose P, Q are two points of U that can be connected by piecewise smooth curve C lying in U. Prove that Vf given that Further if F = where f(x,y) = sin(x — does there exist a smooth, closed path C such that F.dr = 1? If so, find such a path C ii and prove Green’s Theorem for a rectangle Evaluate (3y — esinx ) dx + (7x +1) dy where C is the circle i Define the line integral of a vector field F defined on an open set U in R” along 12 an oriented curve in U. If T and I’ are two equivalent but orientation reversing curves in U, show that F= — F Evaluate + xdy where 8 marks
   • (1) C is the line segment from to (0,2)
   • (2) C is the arc of the parabola x = 4 — y* from (—5,—3) to (0,2)
   • (3) Is F = (y?,x) conservative? Justify your answer Ml the line integral (1 — + dy iv A force field F(x,y) = cxyi y*j, (‘c’ is a positive constant) acts on a particle which moves it from (0,0) to the line x = 1 along a curve of the form y where a>0,b Find a value of ‘a’ (in terms of ‘c’) if the work done by this force is independent of b
   • Q. P. Code : 20499
6. Q3 ai i State and prove Stokes’ theorem for an oriented smooth, simple parameterized surface in bounded by a simple, closed curve traversed counter clockwise it Let S and V satisfy hypothesis of Divergence Theorem, scalar fields f,g have continuous second order partial derivatives and n is the unit outward normal vector to surface S. andr the following 8 marks
   • (2) = =f J, where | V | is volume of V b Evaluate f J,F.ndS where S is the hemisphere above the XY plane with unit 12 it Let S=r(T) be a smooth parametric surface described by a differentiable function r defined on a region T. Let f scalar field and bounded on If R and r are smoothly equivalent parametrizations with R(s,t) = r(G(s,t)) where G(s,t) = u(s,t)i+ v(s,t)j is a one to one continuously differentiable map, then show that fds= fds where G(B) = A ili Use Gauss theorem. to find f where and S is the region by iv Use Stokes’ theorem to evaluate f.x*dx—xydy+z*dz where C is the boundary of the tetrahedron with vertices (2,0,0), (0,2,0) and (0,0,2) lying in the first octant
7. Q4 spherical to evaluate fff, z dxdydz where S is the solid ii Let < y O(x)} where @ is a non negative continuous function on Let f(x,y) be a function on D such that that Jf, f dA =0 iii Let U be an open set in and : [a,b] > U be a parameterization of curve I’ If f > R is a continuous function, then show that f. f = f+ Ir, where I’, and are restrictions of a@ to [a,c] and [c,d] <b iv Evaluate the line integral of f(x,y,z)=x+y+z , along the path v surface area of S, where S is the part of the paraboloid z = x* + y? that lies below the plane If S and C satisfy hypothesis of Stokes’ Theorem and f,g have continuous second order partial derivatives. Then prove that 15 marks

Read from the scan above, so a character or two may differ. The scan is the original.