From the graph below, determine if each integral converges, diverges, or if its behavior cannot be determined.

(a)

(b)

(c)

(d)

(e)

(f)

Answer

(a) Since on (0,1] and

converges, so does

(b) Since on [1,∞) and

diverges, so does

(c) Since

and

diverges, so does the original integral

(d) We can't tell what this function does. The function *g*(*x*) is less than on [0,1), but the integral

diverges, which doesn't tell us anything. The function *g*(*x*) is greater than on [0,1), but

converges, so this doesn't tell us anything either!

(e) We can't tell what

does. Since

converges, it doesn't help to know that on [1,∞).

Since

diverges, it doesn't help to know that on [1,∞).

(f) We can't tell what

does, because we can't tell if either integral on the right-hand side diverges.