Integral #int dx/(x^2sqrt(x^2-16))#?
We will use trigonometric substitution to solve this. Let's set up our triangle:
Now, let's write the basic three trigonometric functions for angle
Let's take the derivative of
Let's plug in for
Let's plug in all the pieces to convert our integral into a trigonometric integral:
Now, we can substitute back:
Find the volume of solid generated from revolving a region bounded by
The graph below shows this area:
If we revolve this area around the
If you can imagine this solid being divided into vertical slices parallel to the
The circles have a radius
So, because the formula for the area of a circle is
Now if we take the integral of this function and evaluate it between
This is because the integral adds the areas of infinite number of discs between the two limits together.
Let's plug this in: