미분적분학실습2-Week-10

912 days ago by jhlee2chn

14.3. Double Integrals in Polar Coordinates

Ex. 1

var('x, y, r, t') f(x, y) = 3*x + 4*y^2 u = r*cos(t) v = r*sin(t) integral(integral(f(u, v)*r, r, 1, 2), t, 0, pi) 
       
15/2*pi
15/2*pi

Ex. 2

var('x, y, r, t') f(x, y) = 1 - x^2 - y^2 u = r*cos(t) v = r*sin(t) integral(integral(f(u, v)*r, r, 0, 1), t, 0, 2*pi) 
       
1/2*pi
1/2*pi

Ex. 3

polar_plot(cos(2*t), (t, -pi/4, pi/4)) 
       
var('r, t') integral(integral(r, r, 0, cos(2*t)), t, -pi/4, pi/4) 
       
1/8*pi
1/8*pi

Ex. 4

var('x, y, z') p1 = plot3d(x^2 + y^2, (x, 0, 2), (y, -1, 1), opacity = 0.4) p2 = implicit_plot3d(x^2 + y^2 == 2*x, (x, 0, 2), (y, -1, 1), (z, 0, 5), color = 'green', opacity = 0.4) p3 = plot3d(0, (x, 0, 2), (y, -1, 1), opacity = 0.4, color = 'orange') p1 + p2 + p3 
       
Sleeping...
If no image appears re-execute the cell. 3-D viewer has been updated.
var('x, y, r, t') f(x, y) = x^2 + y^2 u = r*cos(t) v = r*sin(t) integral(integral(f(u, v)*r, r, 0, 2*cos(t)), t, -pi/2, pi/2) 
       
3/2*pi
3/2*pi

Exercise 1

var('x, y') f1(x, y) = 3*x^2 + 3*y^2 f2(x, y) = 4 - x^2 - y^2 p1 = plot3d(f1, (x, -1, 1), (y, -1, 1), opacity = 0.4) p2 = plot3d(f2, (x, -1, 1), (y, -1, 1), opacity = 0.4, color = 'orange') show(p1 + p2, aspect_ratio = [1, 1, 1]) 
       
Sleeping...
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var('r, t') u = r*cos(t) v = r*sin(t) integral(integral((f2(u, v) - f1(u, v))*r, r, 0, 1), t, 0, 2*pi) 
       
2*pi
2*pi

14.5. Surface Area

Ex. 1

var('x, y') f(x, y) = x^2 + 2*y fx = diff(f(x, y), x) fy = diff(f(x, y), y) integral(integral(sqrt(1 + fx^2 + fy^2), y, 0, x), x, 0, 1) 
       
-5/12*sqrt(5) + 9/4
-5/12*sqrt(5) + 9/4

Ex. 2

var('x, y, r, t') f(x, y) = x^2 + y^2 fx = diff(f(x, y), x) fy = diff(f(x, y), y) u = r*cos(t) v = r*sin(t) integral(integral(sqrt(1 + fx^2 + fy^2).subs(x = u, y = v)*r, r, 0, 3), t, 0, 2*pi) 
       
-1/6*pi + 1/6*(37*sqrt(37))*pi
-1/6*pi + 1/6*(37*sqrt(37))*pi

Exercise 1

var('x, y, z') p1 = implicit_plot3d(x^2 + y^2 + z^2 == 4*z, (x, -2, 2), (y, -2, 2), (z, 0, 4), opacity = 0.3) p2 = plot3d(x^2 + y^2, (x, -2, 2), (y, -2, 2), opacity = 0.3, color = 'red') p1 + p2 
       
Sleeping...
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solve([x^2 + y^2 + z^2 == 4*z, x^2 + y^2 == z], x, y, z) 
       
[[x == r1, y == -sqrt(-r1^2 + 3), z == 3], [x == r2, y == sqrt(-r2^2 +
3), z == 3], [x == r3, y == -I*r3, z == 0], [x == r4, y == I*r4, z ==
0]]
[[x == r1, y == -sqrt(-r1^2 + 3), z == 3], [x == r2, y == sqrt(-r2^2 + 3), z == 3], [x == r3, y == -I*r3, z == 0], [x == r4, y == I*r4, z == 0]]
solve(x^2 + y^2 + z^2 == 4*z, z) 
       
[z == -sqrt(-x^2 - y^2 + 4) + 2, z == sqrt(-x^2 - y^2 + 4) + 2]
[z == -sqrt(-x^2 - y^2 + 4) + 2, z == sqrt(-x^2 - y^2 + 4) + 2]
var('r, t') f(x, y) = sqrt(-x^2 - y^2 + 4) + 2 fx = diff(f(x, y), x) fy = diff(f(x, y), y) u = r*cos(t) v = r*sin(t) integral(integral(sqrt(1 + fx^2 + fy^2).subs(x = u, y = v)*r, r, 0, sqrt(3)), t, 0, 2*pi) 
       
4*pi
4*pi

14.6. Triple Integrals

Ex. 1

var('x, y, z') f(x, y, z) = x*y*z^2 integral(integral(integral(f(x, y, z), x, 0, 1), y, -1, 2), z, 0, 3) 
       
27/4
27/4

Ex. 2

var('x, y, z') f(x, y, z) = z integral(integral(integral(f(x, y, z), z, 0, 1 - x - y), y, 0, 1 - x), x, 0, 1) 
       
1/24
1/24

Ex. 3

var('x, y, z, r, t') f(x, y, z) = sqrt(x^2 + z^2) P(x, z) = integral(f(x, y, z), y, x^2 + z^2, 4) u = r*cos(t) v = r*sin(t) integral(integral((P(u, v)*r).simplify_full(), r, 0, 2), t, 0, 2*pi) 
       
128/15*pi
128/15*pi

Ex. 5

var('x, y, z') integral(integral(integral(1, z, 0, 2 - x - 2*y), y, x/2, 1 - x/2), x, 0, 1) 
       
1/3
1/3

Exercise 1

var('x, y, z') p1 = plot3d(x^2 - 1, (x, -2, 2), (y, -2, 2), opacity = 0.3) p2 = plot3d(1 - x^2, (x, -2, 2), (y, -2, 2), opacity = 0.3, color = 'red') p3 = implicit_plot3d(y == 0, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.3, color = 'green') p4 = implicit_plot3d(y == 2, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.3, color = 'orange') p1 + p2 + p3 + p4 
       
Sleeping...
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solve(x^2 - 1 == 1 - x^2, x) 
       
[x == -1, x == 1]
[x == -1, x == 1]
integral(integral(integral(x - y, z, x^2 - 1, 1 - x^2), y, 0, 2), x, -1, 1) 
       
-16/3
-16/3

Exercise 2

var('x, y, z') p1 = implicit_plot3d(x^2 + z^ 2 == 4, (x, -2, 2), (y, -2, 6), (z, -4, 4), opacity = 0.3) p2 = implicit_plot3d(y == -1, (x, -2, 2), (y, -2, 6), (z, -4, 4), opacity = 0.3, color = 'red') p3 = implicit_plot3d(y + z == 4, (x, -2, 2), (y, -2, 6), (z, -4, 4), opacity = 0.3, color = 'green') show(p1 + p2 + p3, aspect_ratio = [1, 1, 1]) 
       
Sleeping...
If no image appears re-execute the cell. 3-D viewer has been updated.
var('x, y, z, r, t') P(x, z) = integral(1, y, -1, 4 - z) u = r*cos(t) v = r*sin(t) integral(integral((P(u, v)*r).simplify_full(), r, 0, 2), t, 0, 2*pi) 
       
20*pi
20*pi