Math 252 Test # 1 Material Winter 2011 (Test Date: February 4, 2011)

Transcription

1 Math 5 Test # Material Winter (Test Date: Februar, ) EXERCISES Page 5,, 7- Page 9 -, 6, 7 Page 7 -, 6- Page 5,, 8,, Page 6 -, 9-8,, 8 EXERCISES Page 9, 9,, 5. Find the volume above the x-plane bounded b the paraboloid z = x + and the planes x = ±, = ±. [ (x + ) dx] d. Find the volume above the x-plane bounded b the clinder x + = and the plane x++z =. [ x ( x ) x. Find the volume above the x-plane bounded b the clinder = x and the planes = x and z = x +. (x + ) [ x x. Find the volume of the solid bounded b the coordinate planes, the planes x = and = 5, and the surface z = x. [ 5 ) ( x 5. Find the volume above the x-plane bounded b the clinder x + = 9 and the paraboloid z = x +. [ 9 x (( x + 9 x ) ) 6. Find the volume of the solid in the first octant bounded b the clinder = x and the planes x =, z =, =, and x + z =. [ x ( x ) 7. Change the order of integration of (a) [ (b) +x dx]d. [ x [ x+ x +x d]dx xd]dx. [ xdx]d + 9 [ (c) [ x (x + )d]dx. x (x + )dx]d [ (d) [ + 9dx]d. [ x + 9dx]d (e) [ dx]d. [ x x d]dx+ 8 [ x d]dx. (f) π [ cos x d]dx. [ cos dx]d (g) e [ ex dx]d. xd]dx e [ e ex d]dx + [ x e ex d]dx (h) [ ln e x dx]d. ln [ e x d]dx e x

2 (i) [ dx]d. [ x d]dx. (j) π [ x x cos d]dx. π x cos dx]d [ π (k) [ (x + )dx]d. [ x (x + )d]dx + 9 (l) [ (e x )dx]d. [ x (ex )d]dx. (m) [ (x + )dx]d. [ x (x + )d]dx. [ x (x + )d]dx. (n) [ (x + ) dx]d. [ x (x + ) d]dx + [ x (x + ) d]dx. 8. Use double integrals to find the area of the region bounded b the given curves and lines. (a) The parabola x = and the line = x. [ + dx] d (b) The parabola = x x and the line x + =. [ x x x (c) The axes and the line x + = a (a > ). [ a x (d) The -axis, the line = x, and the line = 6. [ 6 x (e) The x-axis, the curve = e x, and the lines x =, x = a (a > ). [ e x (f) The parabolas = x and = x x. [ x x x 9. (a) Evaluate D (x )dxd, where D is the region {(x, ) x, x }. [ x (x ) (b) Evaluate ( sin πx)dxd, where D is the region {(x, ) x, x}. D ( sin πx) [ x (c) Evaluate D (x + )dxd, where D is the region {(x, ), x }. [ (x + ) dx] d (d) Evaluate D (x + )dxd, where D is the region bounded b the positive x-axis, the positive -axis, and the line x + = 9. [ 9 x (x + ) (e) Evaluate D xda, where D is the region {(x, ) x >, x < < x }. 5 [ x x x

3 . Evaluate dxddz, where B is the region bounded b the coordinate planes and the plane B x + + z =. [ x ( x. Find the volume of the sphere x + + z = a. a [ a x a x ( a x a x. Evaluate x x (a) (b) (c) 8x zdzddx x xdzdxd (d) π 5 z 5 (e) π ln x cos ez dzddx π x π ( +z ) xz x + + z dxddz sin dzddx sin dxd =. Evaluate zdxddz, where W is the region bounded b the four planes x =, =, z =, W z =, and the clinder x + =, with x,. [ x ( z. Evaluate W zex+ dxddz, where W = [, ] [, ] [, ]. (e ) 5. Evaluate W (x + + z )dxddz, where W is the region bounded b x + + z = a (a > ), x =, =, and z =. [ a x ( a x (x + + z ) = a5 6. Set up a triple integral for the volume of each of the following solid regions. (a) The region in the first octant bounded above b the clinder z = and ling between the vertical planes given b x + = and x + = [ ( (b) The upper hemisphere given b z = x. [ x ( x x (c) The region bounded below b the paraboloid z = x + and above b the sphere x + +z = 6. [ x ( 6 x x x + 7. Use the triple integration to find the volumes of the given regions. (a) The region in the first octant bounded b the clinder x = and the planes = z, x =, z =. [ ( (b) The region above the x-plane bounded b the surfaces z = 6, z =, = x, =, and x =. [ (

4 (c) The region bounded b the paraboloids z = 8 x and z = x +. [ ( 8 x x + (d) The region bounded b the ellipsoid x b a x a a + b + z c = b b [ a ( c b b c x a b (e) The region bounded b the clinder z = and the paraboloid z = x +. [ ( x + (f) The region bounded b the coordinate planes and the plane x a + b + z c = b [ a( b ) ( x a b (g) The region bounded b the clinder x + = x, the x-plane and the paraboloid z = x +. [ x x ( x x (x + ) 8. Use a double integrals in polar coordinates to find the areas of the indicated regions. (a) The cardioid r = a( + cos θ) (a > ). π [ a(+cos θ) rdr]dθ = πa (b) The circle r = a. π rdr]dθ = πa [ a (c) The circle r = a sin θ. π [ a sin θ rdr]dθ = πa (d) The circle r = a cos θ. π [ a cos θ rdr]dθ = πa π (e) One loop of r = a cos θ (a > ). π [ a cos θ) rdr]dθ = πa 8 (f) One loop of r = cos θ. π 6 [ cos θ) rdr]dθ = π (g) The region inside the lemniscate r = a cos θ and outside the circle r = a (a > ). [ a cos θ a π 6 rdr]dθ = a ( π) (h) The region inside r = tan θ and between θ = and θ = π. π [ tan θ rdr]dθ = 8 ( π) (i) The region inside the cardioid r = a( + cos θ) and outside the circle r = a (a > ). π [ a(+cos θ) rdr]dθ = a a (9 π) (j) The region inside the cardioid r = + cos θ and to the right of the line x =. π [ +cos θ sec θ rdr]dθ = 6 (8π + 9 ) 9. Write the integral in the form β r(θ) α r zrdrdθ. (θ) (a) x zddx. π (b) [ 9 x 9 x zddx. π π [ (c) zdxd. π π [

5 (d) x zddx. x π [ sec θ tan θ (e) (x ) zddx. π [ cos θ (f) zdxd. π [ sin θ 5