Question

Find the volume

The base of S is the region enclosed by the parabola y = 1 - x^2 and the x-axis. Cross-sections perpendicular to the y-axis are squares.

Answer 1

This is just like the shell method, only you're using the surface area of rectangular prisms instead of cylinders.

So the surface area of a rectangular prism is:

A=4wh

I'm making w the width at the bottom, and 4w represents the entire perimeter of the square base, similar to how in the shell method we use the circumference.

Then I'm using h to represent the height, which is represented by the parabola curve.

Now what I would like to do is get this all in terms of x so that I can easily sum up all the rectangles inside each other with respect to x. But first, what are my limits of integration?



It's just where the lines y=1-x^2 and y=0 intersect, so to find that we set them equal to each other. The reason we don't go from -1 to +1 is because the rectangular shell at -1 is the same one at +1, so we'd be counting it twice along with all the others like at -1/2 and +1/2.

Ok, so now back to integrating...
\[V=\int\limits_{0}^{1}(4wh)dx\]
We now need to get everything in terms of x so that we can integrate (sum all the surface areas of rectangular prisms of varying heights).

It would appear that we lucked out on w, We can see from drawing a picture that a point x on the graph is just half of the width, so in math we say 2x=w. Now we need the height. What's the height at any point? It's just the function, y represents the height. So we can jus