Area, Volume, and Arc Length
In the Real World
In this section we left the integrating to the calculators. That's practical, because the calculators can integrate much more quickly and accurately than we can. Instead, we've been spending our time practicing imagination.
The word "imagination" might seem out of place in the "Practical" section of this lesson. For some reason, imagination often gets cast as the villain alongside the hero of practicality. We're not sure why, because imagination is terribly practical.
Imagination means "forming a mental image of something not present to the senses." If you're listening to the sound of the car and trying to envision which part of the engine is broken, that's imagination. If you're preparing to perform surgery and thinking about where the organs lie under the skin, that's imagination. If you're preparing for a race by visualizing yourself running it perfectly, that's imagination.
Imagination means "the act...of forming a mental image of something...never before wholly perceived in reality." If you want to design or make anything, anything at all, you need a working imagination. What are you interested in? Buildings? Cars? Shoes? Three-dimensional puzzles? In order to create something new, you need to be able to see something that doesn't yet exist.
When you practice something, you get better at it. Holding weird 3-D objects and their various slices in your head will make you better at everything else that requires imagination, which is just about everything. Now that's practical.