Please help!

[rjeffries96]

A solid has its base in the xy plane, bounded by the x-axis, y-axis, and the function y=3-(x^5). If cross sections taken perpendicular to the x-axis are semicircles whose diameters are in the xy plane, what is the volume of the solid? ....I'm very confused about cross sections; i was absent the day we learned them and still haven't caught on...please help!

 

[tkhunny]

General Data Dumping

Area of a Circle: \(\displaystyle \pi r^{2}\)

Area of Semi-Circle: \(\displaystyle \frac{1}{2}\pi r^{2}\)

Now to your problem...

Diameter Perpendicular to the x-axis. -- As you slice perpendicular to the x-axis, you begin to see that the diameter of each sliced semi-circle is 'y'. Or, in other symbols, the diameter of each slice is (3 - x^5). Since we need the radius, not the diameter, divide by 2. \(\displaystyle r_{slice} = \frac{1}{2}(3-x^{5})\)

Area of each Semi-Circlular slice, then, is \(\displaystyle \frac{1}{2}\pi r_{slice}^{2}\)

The last challenge is the limits. In this case, it is simple enough. \(\displaystyle [0,\sqrt[5]{3}]\)

Are we getting closer or are we just doing an exploration?