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1- Evaluate (a) 1+4xy dady rsin'e dedr dyda

1- Evaluate (a) 1+4xy dady rsin'e dedr dyda

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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a 1 11:00 Chapter one... Hints Equations of some geometrical shapes Sphare : (x -h)? + (y - k)? + (z - 1)? = a? h, k,l are the center and a is thr radius Ellipsoid : ++2 *ta=1 (a > b>c) y? Paraboloid : z=*7 Cone : z = x + y2 Cylinder : x? + y? = a²for all z x? +z² = a²for all y y? +z? = a²for all x Sheet No (1) 1- Evaluate (a) 1+ 4xy dxdy rsin e dedr 2) dydx 2- Calculate the following double integrals (a) (6x*y - 5y*)dA R= {(x, y)|0 sxs 3,0s ys 1) cos(x + 2y)dA R= {(x,y)|0 < x < 7, 0 S y s (b) (c) xsin(x + y)dA R= [0," - [0, 3 (a) Find the volume of the solid that lies under the plane 3x+ 2y +z = 12 and above rectangular R= (x,y)|o sxs1,0s ys (b) Find the volume of the solid in the first octant bounded by the cylinder z= 16 - x And the plane y=5 (4) Evaluate (a) xy dxdy (b) dydx 5- Evaluate the double integrals (a)" xdA D - ((x, y)|0 < x S R,0sys sinx) (b) D ye""dA D- ((x, y)|0 sys 4,0 sxsy) D 6- Sketch the region of integration and change the order of integration f(x.y) dydx f(x.y)dxdy 7- Find the volume of the solid that lies under the paraboloid z = x² + y? above the xy-plane and inside the cylinder x + y? = 2x 8- Evaluate the triple integrals (a) 6xz dydxdz (b) xsiny dydzdx 9. Use the triple integral to find the volume of the given solid (a) The tetrahedron enclosed by the coordinate planes and plane 2x+ y + z = 4 (b) The solid bounded by the cylinder y = x' and the plane z = 0,7 = 4 and y = 9 10- Evaluate I Vx²+ y² dV where E is the region that lies inside the cylinder E x2 + y? = 16 and between z = -5 and z = 4 11- Find the volume of the solid that lies within both the cylinder x + y? = 1and the sphere x? + y? + z? = 4 (x² + v? + z)?ay where B is a ball with center the origin and B 12- Evaluate гаdius 5. 13-Evaluate I zdy where E lies between the spheres x + y? + z² = 1and E x2 + y? +z? = 4 in the first octant. II
Transcribed Image Text:a 1 11:00 Chapter one... Hints Equations of some geometrical shapes Sphare : (x -h)? + (y - k)? + (z - 1)? = a? h, k,l are the center and a is thr radius Ellipsoid : ++2 *ta=1 (a > b>c) y? Paraboloid : z=*7 Cone : z = x + y2 Cylinder : x? + y? = a²for all z x? +z² = a²for all y y? +z? = a²for all x Sheet No (1) 1- Evaluate (a) 1+ 4xy dxdy rsin e dedr 2) dydx 2- Calculate the following double integrals (a) (6x*y - 5y*)dA R= {(x, y)|0 sxs 3,0s ys 1) cos(x + 2y)dA R= {(x,y)|0 < x < 7, 0 S y s (b) (c) xsin(x + y)dA R= [0," - [0, 3 (a) Find the volume of the solid that lies under the plane 3x+ 2y +z = 12 and above rectangular R= (x,y)|o sxs1,0s ys (b) Find the volume of the solid in the first octant bounded by the cylinder z= 16 - x And the plane y=5 (4) Evaluate (a) xy dxdy (b) dydx 5- Evaluate the double integrals (a)" xdA D - ((x, y)|0 < x S R,0sys sinx) (b) D ye""dA D- ((x, y)|0 sys 4,0 sxsy) D 6- Sketch the region of integration and change the order of integration f(x.y) dydx f(x.y)dxdy 7- Find the volume of the solid that lies under the paraboloid z = x² + y? above the xy-plane and inside the cylinder x + y? = 2x 8- Evaluate the triple integrals (a) 6xz dydxdz (b) xsiny dydzdx 9. Use the triple integral to find the volume of the given solid (a) The tetrahedron enclosed by the coordinate planes and plane 2x+ y + z = 4 (b) The solid bounded by the cylinder y = x' and the plane z = 0,7 = 4 and y = 9 10- Evaluate I Vx²+ y² dV where E is the region that lies inside the cylinder E x2 + y? = 16 and between z = -5 and z = 4 11- Find the volume of the solid that lies within both the cylinder x + y? = 1and the sphere x? + y? + z? = 4 (x² + v? + z)?ay where B is a ball with center the origin and B 12- Evaluate гаdius 5. 13-Evaluate I zdy where E lies between the spheres x + y? + z² = 1and E x2 + y? +z? = 4 in the first octant. II
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