💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!



Numerade Educator



Problem 73 Hard Difficulty

We arrived at Formula 6.3.2, $ V = \displaystyle \int_a^b 2 \pi x f(x) dx $, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where $ f $ is one-to-one and therefore has an inverse function $ g $. Use the figure to show that
$$ V = \pi b^2 d - \pi a^2 c - \int_c^d \pi [g(y)]^2 dy $$
Make the substitution $ y = f(x) $ and then use integration by parts on the resulting integral to prove that
$$ V = \int_a^b 2 \pi x f(x) dx $$


$V=$ $\int_{a}^{b} 2 \pi x f(x) d x$


You must be signed in to discuss.

Video Transcript

problem is we arrived at for Miller Expanse. The Republic, too. Way is they want you. This lines. This is your doing cylindrical shelves. Now we can use interpretation. My most brilliant. Using the slicing My third off of section six point You leased the case. Wired half is one to one and therefore has a humorous function Gene, use a figure to show that we used to go to pine disquiet The mice Hi described seeing minus interior from st today High U Y squire watch and making a substitution. Why goto Max And then use integration. My parts on the resulting integral. To prove that Bea is they got you into girl from a to B two packs. Relax, Tex. Now let's see this graph. No way is the volume when we rotate thiss part I won this part The moment will take his place. A regent about why access then this's athletics and the volume of this week on DH. If we rotate thiss rectangle about y axis, we have volume. This is hi Hey square have seen ifwe don't taint a big rectangle about y axes. I want you. Is this part on the way? Rotate this region reading too about why? Axes in the water. It's this part. So from this graph, it is clear we use the cultural this part of minus this part on minus this part Hard minus. This is too small heart. That's it. Come to this part now if substitution why go to after? Relax then we have expenses equal to T. Why this wise? The humorous function All Blacks now into girl I'm saying to you Hi. Gee, why? Squire? You know why it's cultural. So we have Here is the axis of the two. Do you want the why? This's in cultural integral from state to state. Hi, I'm exploring terms. So here. So here we don't need just one is wise He goto earthworks. You want to two half flux from the ex This is he going with this one and then use substitution used integration Pars become light you is expired and the problem is they tow truck. Then you promise to lax The three is equal to I've axe and this integral is you go to when he was a farm owner of into question my parts But this isn't going to you house. We this is I score. This is pi times I scored square pounds off works from See Teo Now see here. One way. Use this substitution if. Why? From sea to Dean. Access from a to B. This is and to be here. This is from a minus into grow your crown Ma'am sleeves this his two lives for life? Yes, from I need to be. But this is e cultural. Hi. Esquire is the minus. Hi. A squire a C minus into grow a to B too high x Lex Jax, this is Auntie Girl on this part. So Wei is cultural. I be squired from Stevie minus by a squire And see my ass this light just hide d squared times Hi. A square times my justice two hi's life sticks. This's exactly this one.