CoCalc -- 2021-12-08-gouvea-integrate.ipynb

Path: support2/2021-12-08-gouvea-integrate.ipynb

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Image: ubuntu2004-dev

Kernel: SageMath 9.4

I was showing my students a famous calculus example of an integral that can be computed in one order of the variables but not in the other. Knowing that SageMath can compute anything, the students suggested trying the integral the "wrong" way. The "right" way is

In [1]:

var('x,y')
right = integrate(integrate(sin(x^2),y,0,x),x,0,1); right

-1/2*cos(1) + 1/2

The "wrong" way is

In [2]:

wrong = integrate(integrate(sin(x^2),x,y,1),y,0,1); wrong

-1/16*(-1)^(3/4)*((sqrt(2) + 4*(-1)^(1/4))*e^I - sqrt(-I)*((I + 1)*sqrt(2)*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2)*(-1)^(1/4)*e^I) + I*sqrt(2)*e^I - 2*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2) - 2*(-1)^(1/4))*e^(-I)

In [3]:

wrong == right

-1/16*(-1)^(3/4)*((sqrt(2) + 4*(-1)^(1/4))*e^I - sqrt(-I)*((I + 1)*sqrt(2)*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2)*(-1)^(1/4)*e^I) + I*sqrt(2)*e^I - 2*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2) - 2*(-1)^(1/4))*e^(-I) == -1/2*cos(1) + 1/2

In [5]:

# In Sage this means that sage couldn't prove they are equal
bool(wrong==right)

False

In [6]:

CC(wrong)

0.229848847065930 - 4.16333634234434e-17*I

In [7]:

CC(right)

0.229848847065930

In [8]:

bool(wrong.real() == right)

True

In [9]:

wrong.imag()

0

In [0]: