Work it should be clear that 1 e x x 1 x and we know

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Work: It should be clear that 1 + e - x x > 1 x and we know that Z 1 1 x d x is divergent, so our integral Z 1 1 + e - x x d x is divergent by using the comparison theorem. 4 Examples 1. Show that Z 1 1 x d x is divergent. 2. Show Z 0 1 1 + x 2 d x = π 2 4
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3. Show Z 0 1 e x d x = 1 4. Show Z 1 1 - x e x d x = - 1 e 5. Show Z -∞ e x 1 + e 2 x d x = π 2 6. Show that Z 2 0 1 x 3 d x is divergent. 5
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7. Show Z 0 1 x ( x + 1) d x = π. 8. The solid formed by revolving the region between 1 /x and the x -axis, for x 1 is called Gabriel’s Horn. What is most weird about this object is that it can be shown to have finite volume, but its surface area is infinite. Set-up and evaluate the integral for its volume and verify that it is π . 9. Show that Z 0 1 e x 2 d x is convergent. 8 8 Hint: notice that e - x 2 e - x , look over the past problems, and use the comparison theorem. 6
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