### Video Transcript

The curve shown is ๐ฆ equals one
over ๐ฅ. What is the area of the shaded
region? Give an exact answer.

Now we recall that the area of the
region bounded by the curve ๐ฆ equals ๐ of ๐ฅ, the lines ๐ฅ equals ๐ and ๐ฅ equals
๐, and the ๐ฅ-axis is given by the integral from ๐ to ๐ of ๐ of ๐ฅ with respect
to ๐ฅ. In this case, weโre told that the
function ๐ of ๐ฅ is one over ๐ฅ. And from the graph, we can see that
the values of the limits for this integral are negative one for the lower limit and
negative one-third for the upper limit. So we have the definite integral
from negative one to negative one-third of one over ๐ฅ with respect to ๐ฅ.

We then recall that the integral of
one over ๐ฅ with respect to ๐ฅ is equal to the natural logarithm of the absolute
value of ๐ฅ plus the constant of integration. And that absolute value is really
clear here because the two values for our limits are both negative. And we recall that the natural
logarithm of a negative value is undefined. So we must make sure we include
those absolute value signs.

So weโre taking the natural
logarithm of a positive value. We donโt need the constant of
integration ๐ here as weโre performing a definite integral. Substituting the limits gives the
natural logarithm of the absolute value of negative one-third minus the natural
logarithm of the absolute value of negative one. Thatโs the natural logarithm of
one-third minus the natural logarithm of one. And at this point, we can recall
that the natural logarithm of one is just zero. So our answer has simplified to the
natural logarithm of one-third.

Now this may not be immediately
obvious to you. But in fact, the natural logarithm
of one-third is a negative value. We can see this if we recall one of
our laws of logarithms, which is that the logarithm of ๐ over ๐ is equal to the
logarithm of ๐ minus the logarithm of ๐. And so the natural logarithm of
one-third is the natural logarithm of one minus the natural logarithm of three. And again, we recall that the
natural logarithm of one is equal to zero. So our answer appears to be that
this area is equal to negative the natural logarithm of three.

This doesnโt really make sense
though as areas should be positive. What we see then is that when we
use integration to evaluate an area below the ๐ฅ-axis, we will get a negative
result. This doesnโt mean though that the
area itself is negative. The negative sign is just
signifying to us that the area is below the ๐ฅ-axis.

Really then, what we shouldโve done
is include absolute value signs around our integral sign at the beginning. And what this means is that
although the value of the integral is negative, the natural logarithm of three, the
value of the area is the absolute value of this. So thatโs just the natural
logarithm of three. The integral is negative to signify
that the area is below the ๐ฅ-axis. But the area itself is
positive. So our answer to the question, and
it is an exact answer, is that this area is equal to the natural logarithm of three.