Find the Error: U-Substitution

It is a beautiful Spring morning. You and your friends are out on the beach. There are volleyball nets up, several open barbecue pits and picnic tables, friendly and interesting people waiting to meet people just like you, and a cabana offering free hot-dogs and broccoli bites. You are all sitting in a circle, practicing integration by u-substitution. You feel a presence behind you, and see a wild-eyed, hungry-looking stranger staring down.

“What are you kids doing lollygagging around? Don’t you have any work to do?”

“We are practicing integration by u-substitution,” you explain. “One of us does an integral, and the others check to see if it was done right.”

“Hmm.” he says. Suddenly he gets an evil twinkle in his eyes. “Maybe I had you kids figured all wrong. Mind if I try?”

You are suspicious. When you’ve seen him before, he has been quite unfriendly to both you and the subject of Calculus in general. But maybe he has finally realized how useful that Mathematics really is. He writes the following in the sand.

“Now I swear by Eisenhower’s Earwax that the answer is therefore zero but maybe you can tell me for sure.” The stranger is looking directly at you when he says this. Your friends are staring, too.

You go through it line by line. “It looks okay to me, and the integral from 1 to 1 of anything is zero.”

“Funny enough,” he replies, “because last I heard the graph of 1/(1+x4) looks like this:

And the area under that puppy is certainly not zero! What do you think of THAT?”

Your friends all smile condescendingly at him, waiting for you to point out the flaw in his proof. Will you let them down? Have you shown the concept of area is useless? Is there really such a thing as a two-by-four? Or can you find what went wrong with the stranger’s reasoning.

With a tip o’ the hat to Mary Harrison for correcting the UNintentional errors!

  1. Find the error: Differentiation
  2. Find the error: L’Hopital’s Rule
  3. Find the error: Related Rates
  4. Find the error: Fundamental Theorem of Calculus
  5. Find the error: U-substitution
  6. Find the error: Integration by Parts (part 1)
  7. Find the error: Integration by Parts (part 2)
  8. Find the error: Trigonometric Integration
  9. Find the error: Taylor Series
  10. Find the error: Improper Integrals and Taylor Series
  11. Find the error: Separation of Variables
  12. Find the error: Separation of Variables – Exponential Growth
  13. Find the error: Solutions
  14. Find the error: Credits