Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln(6x), y = 4, y = 6, x = 0; about the y-axis Step 1 We will use horizontal strips. Such a strip has a vertical span of Δy. Sketch the region.

Answer 1

Answer:

V = 2219.08π

Step-by-step explanation:

If you use the washer/disk method as you indicated you were going to in the statement "we will us horizontal strips", you have to solve that equation for x.  The equation solved for x is

$x=\frac{1}{6}e^y$

The bounds are y-intervals from 4 to 6, and we are rotating about the y-axis.  I cannot sketch the graph for you here, but if you need it, graph y = ln(6x) and you'll see what it looks like.  I can only help you with the calculus of it, not the drawing of it.

The volume formula then for this is

$V=\pi \int\limits^6_4 {(\frac{1}{6}e^y)^2-(0)^2 } \, dy$

Squaring what's inside the parenthesis gives you:

$V=\pi \int\limits^6_4 {\frac{1}{36}e^{2y} } \, dy$

To make this a tiny bit simpler we can pull out the fraction:

$V=\frac{\pi}{36}\int\limits^6_4 {e^{2y}} \, dy$

The antiderivative of e to the 2y power is found:

$V=\frac{\pi}{36}[\frac{1}{2}e^{2y}]$ from 4 to 6

We can then pull out the 1/2:

$V=\frac{\pi}{72}[e^{2y}]$ from 4 to 6.

Evaluating that antiderivative using the First Fundamental Theorem of calculus gives you:

$V=\frac{\pi}{72}[e^{12}-e^8]$ which simplifies to

$V=\frac{\pi}{72}(159773.8334)$

If you leave your answer in terms of pi it is:

2219.08π or if you multiply pi in it is:

6971.44863