The curve shown is 𝑦 equals one
over 𝑥. 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 𝑦 equals 𝑓 of 𝑥, the lines 𝑥 equals 𝑎 and 𝑥 equals
𝑏, and the 𝑥-axis is given by the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 with respect
to 𝑥. In this case, we’re told that the
function 𝑓 of 𝑥 is one over 𝑥. 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 𝑥 with respect to 𝑥.
We then recall that the integral of
one over 𝑥 with respect to 𝑥 is equal to the natural logarithm of the absolute
value of 𝑥 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 we’re taking the natural
logarithm of a positive value. We don’t need the constant of
integration 𝑐 here as we’re 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. That’s 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 𝑎 over 𝑏 is equal to the
logarithm of 𝑎 minus the logarithm of 𝑏. 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 doesn’t 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 𝑥-axis, we will get a negative
result. This doesn’t mean though that the
area itself is negative. The negative sign is just
signifying to us that the area is below the 𝑥-axis.
Really then, what we should’ve 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 that’s just the natural
logarithm of three. The integral is negative to signify
that the area is below the 𝑥-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.