I tried finding this question on this forum, and also searched it on Google, but did not come up with any credible answers, as a last resort I decided to post this question after pondering on it for a while and not being able to figure it out myself. My apologies if it already exists.
Question: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. (Stewart Calculus 8th Edition, Chapter 6, Section 6.2, Page 446, question 11).
$ y = x^2$ and $x=y^2 $ about $ y = 1$
My understanding: Since we have established that $ y = x^2 $ it means that $ {\sqrt y} = x $. Then we use the washer method to find the area function of the washer and then solve the $\int_a^b A(x)dx$ function. My answer is negative (the correct answer is positive of what I get). I have been informed that I am supposed to work with $ y = x^2$ and $ y = {\sqrt x}$ instead of working with $x = {\sqrt y} = x$ and $ x = y^2$. I am failing to understand that why would this lead to my answer being negative? The values are equal, if they are equal then shouldn't my answer be positive? Why does it matter that I specifically use $ y = x^2$ and $ y = {\sqrt x}$? If the values are equal my answer should just end up being positive. Can anyone here explain to me what am I missing in my thinking?
I understand the calculations and the theory behind integrals. I am specifically concerned about the negative answer and how big of a difference it ends up making if I choose $y$ as my variable rather than $x$.