let's recall the vertical line test, if a vertical line hits the graph or points twice, then is NOT a function. Check the picture below.
A very easy way to find out if a graph is a function is to use the vertical line test.
Look at each graph and mentally draw a vertical line that goes through each of its points.
If that line goes through more than one point (2, 3, 4...) then that graph is not a function.
That's because a function, is an equation for which each input (x value) has exactly one output (y value). There cannot be two points of the same x value, with two different y values.
The two curves intersect at two points, (0, 0) and (8, 8):
The area of a semicircle with diameter is
. The diameter of each cross-section is determined by the vertical distance between the two curves for any given value of
between 0 and 8. Over this interval,
and
, so the volume of this solid is given by the integral