mth.122.handout.09

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

• Notes
• 6

This preview shows pages 4–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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
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
This is the end of the preview. Sign up to access the rest of the document.
• Spring '10
• Ban

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern