Consider the following four integrals: Which of these integrals converge? (Hint:

Question

Consider the following four integrals: Which of these integrals converge? (Hint: compare to "pure" powers of x.) Compute the exact value of at least one of the convergent integrals. Suppose that a is a positive constant and that R is the region bounded above by y = 1 /xa, below by y = 0. and on the left by the line x = 1. Sketch the curves y = 1/xa for a = .5, 1 and 2. Which of these is closest to the x-axis? For which positive numbers a do you get a convergent integral when you attempt to calculate the area of R? Same as b), but for the volume of the solid obtained by rotating R around the x-axis. Same as c). but for the volume of the solid obtained by rotating R around the y-axis.

Explanation / Answer

c and d diverges for sure not the a and b converges as 1+x^4>x^4 so x/(1+x^4 c and d diverges as d is 1-1/(1+x^4) the integral and c is dz/z z varies from 1 to infinity z=1+x^4

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