Great "thinking outside the box" calculus example





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College Discussion Forums: SAT/ACT Tests and Test Preparation: August 2003 Archive: Great "thinking outside the box" calculus example
By Fairyofwind (Fairyofwind) on Thursday, August 14, 2003 - 10:19 pm: Edit

Find the value of this definite integral:

Integral[Sin[x^2]/Sin[x],x,-1,1].

By Jason817 (Jason817) on Thursday, August 14, 2003 - 10:43 pm: Edit

is it .919??? My calc comes up with 0 but you're looking for the positive area of the shaded region.

By Fairyofwind (Fairyofwind) on Thursday, August 14, 2003 - 10:44 pm: Edit

No it's 0. But show it WITHOUT using your calculator!

By Serene (Serene) on Thursday, August 14, 2003 - 10:47 pm: Edit

Nice fairy! =)

One of our calc class tshirt designs (eventually rejected) was to have a really really complicated integral that go from -pi to +pi, just like that =)

By Jason817 (Jason817) on Thursday, August 14, 2003 - 10:51 pm: Edit

Well any definite integral like that involving a function with origin symmetry and boundries of -x and +x would end up being zero, will it not?

By Fairyofwind (Fairyofwind) on Thursday, August 14, 2003 - 10:55 pm: Edit

Hehehehe... good! But because of the way the improper integral is defined, an integral from say -2003 to 2003 of that is not 0, due to infinities where Sin[x]=0, but otherwise yes!

By Serene (Serene) on Thursday, August 14, 2003 - 10:57 pm: Edit

Jason: not all definite integrals. For example,
ò-x x 1/sin(x) dx will not get you 0 =) Gotta be careful!

Yeah. Fairy was careful to include sin(x^2), so there's a limit at x=0 =)

By Jason817 (Jason817) on Thursday, August 14, 2003 - 11:02 pm: Edit

Ah yes, good point.


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