15.1-4 - Integrals of Multivariable Functions.sagews

var('x,y')

(x, y)

f = 45 - x^2 - y^2
plot3d(f, (x,-6,6), (y,-3,3), aspect_ratio=[1,1,0.25])

3D rendering not yet implemented

I1 = integral(f, x, -6, 6)
I1

-12*y^2 + 396

integral(I1, y, -3, 3)

2160

integral( integral(f, x, -6, 6), y, -3, 3)

2160

integral(integral(f, y, -3, 3), x, -6, 6)

2160

var('x,y')
f = 9 - x + 2*y
I1 = integral(f, y, 0, 1/2*x)
I1

(x, y) -1/4*x^2 + 9/2*x

Ans1 = integral(I1, x, 0, 1)
Ans1

13/6

I1 = integral(f, y, x-1, 1/2*x)
Ans2 = integral(I1, x, 1, 2)
Ans2

13/6

integral( integral(f, x, 2*y, y+1), y, 0, 1)

13/3

integral(integral(3*x - 2*y, y, x^3, 8), x, 0, 2)

-2832/35

var('x,y')

(x, y)

I1 = integral(3*x - 2*y, x, 0, y^(1/3))
I1

-2*y^(4/3) + 3/2*y^(2/3)

integral( I1, y, 0, 8)

-2832/35

integral(integral(3*x - 2*y, x, 0, y^(1/3)), y, 0, 8)

-2832/35

var('x,y')

(x, y)

f = 3*y
plot3d(f, (x,0,1), (y,-1,1))

3D rendering not yet implemented

## Polar plot of cardioid and circle
C = polar_plot( 1 + cos(x), (x, 0, 2*pi), color='black')
S = polar_plot( 1, (x, 0, 2*pi))
C+S

C2 = polar_plot( 1 + cos(x), (x, -pi/2, pi/2), color='black')
S2 = polar_plot( 1, (x, -pi/2, pi/2))
C2+S2

var('r')
integral(integral(r, r, 1, 1+cos(x)), x, -pi/2, pi/2)

r 1/4*pi + 2

## Polar plot of lemniscate
P1 = polar_plot( 2*sqrt(cos(2*x)), (x, -pi/4, pi/4))
P2 = polar_plot( -2*sqrt(cos(2*x)), (x, -pi/4, pi/4))
P1+P2