OU BSc Computer Science 2021 Important Question Bank || Osmania University (Important Questions)
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1S
Subject Title: Integral Calculus Prepared by: Ms Afreen Begum
Year: 3 Semester: 5 Updated on: 30.12.2020
Unit - I AREAS AND VOLUMES
- Definition of Double integrals, Riemann sum, Integral over a Rectangle, Properties of the integrals
- Integrate (i) ?2 ?3(x² + y)dydx (ii) ?°/2 ?0 xcosy dx dy
- Compute ?0¹ ?y cosxzdx dy , by changing the order of integration
- State Fubini's theorem.
- Change the order of integration and evaluate ?0¹ ?x sinx dydx + ?1 ?p sinx dydx
- Let R=[-3,3]x[-2,2]. Without explicitly evaluate ?R(x5 + 2y)
- Find the area of the region ,using double integrals, bounded by y=2-x² and x-y=0,2x+y=0
- Integrate the function f(x, y)=3xy over the region bounded by y=2x³ and y=vx.
- Evaluate ?D (x — 2y)dA, where D is the region bounded by y=x²+2 and y=2x³-2.
- Use double integrals to find the area of the region bounded by the parabola y=2-x², and the lines x-y=0, 2x+y=0.
- Find the volume of the region under the graph of f(x,y)=2-|x|-|y| and above the xy-plane .
- Change of order of integration and evaluate (i)?0¹ ?0? (2 — x — y)dydx (ii) ?0² ?04-y² 5xy dy dx (iii) ?0² ?y/2 54/y x dx dy.
- Find the volume of the region bounded by a graph of f(x,y)=2x²+y²sinpx on top ,the xy-plane on the bottom and the planes x=0,x=1,y=-1,y=2 on the sides.
- Compute ?0¹ ?0¹ (5 — |y|)dxdy
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- Discuss the types of elementary regions in the plane used in evaluating the double integrals.
Unit-II TRIPLE INTEGRALS
- Define Integral over a box, Riemann sum of triple integrals, triple integrals, Fubini’s theorem of triple integrals, Elementary region in space, types of triple integrals.
- Find the volume inside the capsule bounded by the paraboloids z=9-x²-y² and z=3x²+3y²-16.
- Evaluate the triple integrals (i) ?[1,1]x[0,2]x[1,3] yvz dV (ii) ?[1,e]x[1,e]x[1,e] xyz dV
- Evaluate ?0¹ ?1+y ?z+z x dx dz dy.
- Integrate the given function over the indicated region W (i) f(x,y,z)=2x-y+z, W is the region bounded by the cylinder z=y², the xy-plane and the planes x=0, x=1, y=-2, y=2. (ii) f(x,y,z)= 8xyz; W is the region bounded by the cylinder y=x², plane y+z=9 and the xy- plane.
- Change the order of integration of ?0¹?0¹?0?+y+z f(x, y, z)dz dxdy to give the five other equivalent iterated integrals.
- Find the volume of the solid bounded by z=4-x², x+y=2 and the coordinate planes.
- Find the volume of the solid bounded by the planes y=0, z=0, 2y+z=6 and the cylinder x²+y²=9.
- Find the volume of the solid bounded by the paraboloid z=4x²+y² and the cylinder y²+z=2.
- Find the volume of the solid over the function f(x,y,z)=4x+y and W is the region bounded by x= y²,y=z, x=y and z=0
- Find the volume of the ellipsoid x²/a² + y²/b² + z²/c² = 1.
- Evaluate ?0¹ ?0¹ ?0¹ (x + 2y + z)dxdydz.
- Integrate the function f(x,y,z)=x+y over the region bounded by x² + 3z = 9 and y=0 , x+y=3.
- Compute ?W 3xdv where W is the region in the first octant bounded by z=x² + y² ,x=0 ,y=0 and z=4.
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- Evaluate ?B f(x,y, z)dv where B is the tetrahedron with the vertices (0,0,0),(1,0,0),(0,1,0) and (0,0,1) and f(x,y,z)=1+xy.
Unit-III CHANGE OF VARIABLES
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- Change of variables and coordinate transformation, Jacobian of double and triple integrals.
- Evaluate ?D (x² — y²)exy dxdy, where D is the region in the first quadrant bounded by the hyperbolas xy=1, xy=4 and the lines y=x, y= x+2.
- Double integrals in polar coordinates, Cartesian coordinates, general coordinates Change of variables in triple integrals, triple integrals in cylindrical coordinates, spherical coordinates.
- Calculate the volume of the cone of height ‘h’ and radius ‘a’, in which the cone is a solid W bounded by the surface az = hv(x² + y²) and the plane z=h. (using both cylindrical and spherical coordinates).
- If T(u,v)=(3u,-v),find the matrix A such that T(u,v)=A(u,v). Describe how T transforms the unit square [0,1]x[0,1]?
- Suppose T(u,v)=(u+v, u-v). If T(u,v,w)=(3u-v,u-v+2w,5u+3v-w),describe how T transforms the unit cube [0,1]x[0,1]x[0,1]?
- Evaluate the integral ?0² ?0² (2x —y)dxdy by using substitutions u=2x-y , v=y.
- Determine the value of ?W (x +y + z)dv where W denotes the solid region in the first octant between the sphere x² + y² + z² = a² and x² + y² + z² = b² where 0
- If T(u,v)=(u-v, 2u+3v) and D* is the parallelogram whose vertices are (0,0),(1,3),(-1,2) and (0,5). Determine D=T(D*).
- Find the area of the region inside both of the circle r=2a cos? and r=2a sin? where a is a positive constant.
- Find the volume of a ball of radius a using spherical coordinates.
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- Find the volume of a cone of radius a and height h using spherical coordinates.
- Evaluate ?3³ ?v(9-x²)v(9-x²) ?0v(9-x²-y²) dzdydx by using cylindrical coordinates.
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