Investigating Ratios of Areas and Volumes Speech or Presentation

Investigating Ratios of Areas and Volumes - Speech or Presentation ExampleA in writing(p) depiction of the said arenas is shown belowKnowing the value of the area B, the area A may also be computed. Instead of integrating the function, it is simpler to subtract the area below the slew from the total area of the rectangle. This results to area A as shownBy comparing the plots of two curves, it can be seen that the area under the curve decreases as the exponent increases. Consequently, the proportion of the two areas also increases. This trend supports the calculated data.This shows that for the section of the curve from x = 0 to 1, the ratio is just equal to the exponent. However, does this possibility hold true for all ranges? To examine this, we reevaluate the generic function y = xn with various rangesWith a range of 0 to 2 for x, the ratio of the areas ashes equal to n. However, only the upper limit has been changed in this particular case. By adjusting the lower limit, a cer tain area is removed from the rectangle as depicted in the following graph.Again, the resultant ratio is simply equal to the exponent n. In fact, it can be proven that the ratio leave remain constant disdain any changes in the limit. With a lower limit of a and an upper limit of b, the proof will appear as shown.Now that it is clear that the ratio of the two areas will remain constant, it is possible to extend the analysis to a three-dimensional one by revolving the surface around a specified axis. First, we will investigate the effects of a revolution around the x-axis.Since the area being revolved is that of B, the resulting volume is taken by evaluating the previous expression. The radius in such(prenominal) a case would be equivalent to y or generally xn. Using this, the volume can be determined in a univocal manner.The remaining volume is that of the revolution of surface A. However, instead of integrating, it is