The Function and its Double-Integral

The function given in section 13.1 #12 is:

> f:=(x,y)->cos(x^4+y^4);

We want to evaluate the double integral of this function over the square region R = [0,1] x [0,1].

Here's what the function looks like in the region R:

> plot3d(f(x,y),x=0..1,y=0..1,axes=frame,orientation=[70,75]);

Let us start by finding the indefinite integral:

> int(int(f(x,y),x),y);

Whoah! What a mess!.

Let's find the double integral over the square region R:

> int(int(f(x,y),x=0..1),y=0..1);

Maple just spit the integral back at us! Strange, how it can do the indefinite integral but not the definite integral.

Let's try Maple's numerical routines:

> st:=time():
intf:=evalf(int(int(f(x,y),x=0..1),y=0..1));
time()-st;

It took Maple 27.45 seconds to come up with this answer on my computer.

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