The full question is in the title, please help using any method
in Calculus Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

x=e^y², y=√ln(x). This curve meets the line y=1 when 1=√ln(x), that is when x=e.

The solid generated by revolution can be split into the volume of a unit cylinder (height and radius=1), which  is π cubic units (x between 0 and 1), plus the difference between the volume of a cylinder with radius 1 and length e, which is πe cubic units, and π∫y²dx, for x between 1 and e.

So we have volume=π+πe-π∫y²dx[1,e].

Substitute ln(x) for y², we need to integrate ln(x)dx, so let dv=dx and u=ln(x), so v=x and du=dx/x. Integrating by parts we have uv-∫vdu=xln(x)-∫dx=xln(x)-x=x(ln(x)-1). When x=e this evaluates to 0, and when x=1 it evaluates to -1. So π∫y²dx=(0-(-1))π=π.

Therefore the volume is π+πe-π=πe cubic units.

by Top Rated User (764k points)

Related questions

Welcome to, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
85,099 questions
90,236 answers
60,405 users