The definite integral is a useful tool in calculus for finding the area between a function and the x-axis. In particular, the area under the parabola y = x^2 can be found using the fundamental theorem of calculus, but there is also a geometric approach using Cavalieri's principle that relies only on pre-calculus concepts. By extruding the region into a prism and then a pyramid, it is shown that the area under the parabola is equal to one-third x^3, with appropriate dimensional units. This approach can also be extended to find the area function for other power functions.
Sure, here are the key facts extracted from the text:
1. The definite integral is a basic tool of calculus.
2. It allows finding the total area between a function and the x-axis between any two points.
3. An example is finding the area under the parabola y equals x squared.
4. The integral from 0 to 3 of x squared DX equals 9.
5. The area between 0 and any point along the x-axis can be represented as the integral from 0 to X of t squared DT.
6. The area function for the parabola y equals x squared measured from zero is 1/3 x cubed.
7. The author wants to find the area under the parabola without using calculus tools.
8. The author uses geometric concepts to solve the problem.
9. Cavalieri's principle is used to equate a volume to an area.
10. The final result for the area under the parabola y equals x squared is one-third X cubed.
11. The author discusses dimensional consistency in the calculations.