Calculus All-in-One For Dummies (+ Chapter Quizzes Online)

Chapter 18

Taming the Infinite with Improper Integrals

IN THIS CHAPTER

The hospital rule — in case studying calculus makes you ill

Meeting integrals without manners

The paradox of Gabriel’s horn

The main topic of this chapter is really amazing when you stop and think about it: calculating the area (or volume) of shapes that are infinitely long. The word infinity comes up in mathematics so often that perhaps we become jaded about the concept and forget how truly incredible it is. It’s about 93 million miles from the Earth to the Sun. That distance is so great that it’s nearly impossible to wrap our minds around it, but it’s nothing compared to the distance to Alpha Centauri A (the nearest star), which is 4.24 light-years away — about 268,000 times as far as the distance to the Sun. Our Milky Way Galaxy is about 100,000 light-years across, and it’s about 4.5 million light-years to our nearest spiral galaxy neighbor, the Andromeda Galaxy. Go out about 10,000 times that far and you reach the “edge” of the observable universe at about 46 or 47 billion light-years away. That’s definitely quite a ways out there, but it’s nothing compared to infinity.

The shapes you deal with in this chapter ...