Posted by Rain on Sunday, September 10, 2006 at 1:45am.
a) Is ∫[-1 to 1]e^x^3 dx positive, negative, or zero? Explain.
I think is positive but i don't know how to explain it.
b) Explain why 0 < ∫[0 to 1] e^x^2 dx <3.
Can you type this equation or whatever in a better format? What branch of math does this problem come from?
Rachel
It's definitely a calculus problem, but some of the symbols didn't translate to ASCII. You'll need to give a brief description of the problem for the symbols you can't type.
Is a) the integral of e^(x^3)dx? on the interval [-1,1]? If so, then you should see that the function is positive on that interval, thus the area is positive.
Part b) appears to put a bound on the definite integral e^(x^2) on the interval [0,1]. f(x)=e^(x^2) is a monotonic increasing funtion on R, has a min of 1 on that interval, and a max of e on that interval The area is <=e<3, and the area is positive since the function is positive there. Thus 0<f(x)<3 for x in [0,1]
After checking my work I see I stated the function for b) wrong. f(x) is monotonic increasing on the interval [0,1], not on R.
Both problems are asking you to make determinations about the area of some exponential function. In b) e^(x^2) is symmetric about the x-axis, but it's increasing on the interval [0,1]. To see this, simply differentiate the function and you'll see it's slope is increasing at every point there.
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