In 6.1, we learned about the area between two curves, but now, Mr. French and the James Stewart have decided to make it more interesting by introducing a third dimension. We are now figuring out the volume between two curves, or the volume caused by the rotation of the area between two curves.
So, to calculate the volume we would use an infinite amount of infinitely thin boxes to calculate it, like a Riemann sum. Because it's a square we know that:And the formula for a circle is: where R is the radius. So the set up would look like this:
The green side, S, is double what the y value is for that x value, which means thatUsing this information, we can now define A in terms of x, like in the general equation for the volume, and find the volume in terms of R, which is an undefined constant.
Using the equation for the circle, we get that:And according to the general equation for volume:
So,Since the circle is also symmetrical about the y-axis, taking the integral from -R to R is the same thing as doubling the integral of 0 to R.Now, we just do what we know how to do - evaluate the integral, keeping in mind that R is a constant.And we're done!
There's another type of volume that Mr. French and James Stewart might ask you to find, and that's a rotation volume. It's when you take an area and rotate it around an axis, creating a solid with circular cross-sections. When the base of the area touches the rotation axis, it creates a disc volume, and when it doesn't, it creates a washer volume.
Take for instance the graph of the square root of x from 2 to 9.And let's rotate it around the x-axis. And since the base of the area touches the axis of rotation, we have a disc volume, and because the cross sections are circles, the integral for the volume would be:
Just as a recap, Jack, Kate and Sawyer captured by the Others, and Jack had Benry on the table in the middle of back surgery, when he insisted that Kate and Sawyer be let go or he'd leave Benry on the table to die. So that should make for an interesting episode tonight!