Practice question · Multiple choice
The "split into thin pieces" argument gives the right arc length with but the wrong surface area if you use instead. What fails?
Hints
- Approximate a diagonal line by horizontal steps and measure the staircase.
- What does the staircase converge to in shape, and in length?
Show the answer
B. The piece is no longer nearly simple where it matters
Why
The staircase paradox: the shape converges and the length does not. Additivity is not enough, the piece must be nearly simple in the quantity being measured, which is why and not appears.
Practise Introduction to Differential Equations
The app has 5 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.
More questions on Introduction to Differential Equations
- Order the steps for solving a first-order separable differential equation with an initial condition.
- The order of a differential equation determines how many independent constants its general solution carries.
- Arc length, volume, work and centre of mass all end up as integrals. What do these quantities share that…
- The average value of f on [a,b] is (1/(b−a))∫f. Why is that the right definition, when an average is normally…