We want to sketch the region enclosed by y equals e raised to the power of 3x and
y equals e raised to the power of 4x and x equals 1. Then we want to find the area of the region.
So if we graph these on the coordinate plane, the graph looks like this. We should recognize
that this blue function here would be e raised to the power of 4x because notice as x increases,
e raised to the power of 4x increases faster than e raised to the power of 3x. Notice on the left,
they intersect when x equals 0, and on the right we're told x equals 1.
So we're looking for the area of this shaded region here.
It is important to identify which function is on top and which function is on the bottom because
when finding the area between two functions, we want to find the integral of f of x minus
g of x from a to b, where f of x will be the top function and g of x will be the bottom function.
So we're finding the area below the top function and above the x-axis and then we're subtracting
the area below the bottom function and above the x-axis, leaving us with the area between
the two functions where we start integrating at x equals a and stop integrating at x equals b.
So looking at our graph, notice how the lower limit of integration would be zero,
so a equals 0, and b would be equal to 1, the upper limit of integration. Therefore, the area
is going to be equal to the integral of, again, the top function minus the bottom function, which
should be e raised to the power of 4x minus e raised to the power of 3x integrated from 0 to 1.
Let's evaluate this. On the previous slide,
notice to find the anti-derivative of e raised to the power of 4x,
we have to perform a u substitution where u is equal to 4x, and therefore, the differential u
is equal to 4 dx. So if we divide both sides by 4, notice that 1 over 4 du is equal to dx.
So integrating this, we'll have an extra factor of 1 over 4, which means this will be equal
to 1 over 4 e raised to the power of 4x minus. When we integrate e raised to the power of 3x,
we also have to perform a substitution. Let's call it v, so if we let v equal 3x,
notice dv is equal to 3 dx. Dividing both sides by 3, we have 1 over 3 dv equals dx.
So integrating e raised to the power of 3x, we have an extra factor of 1 over 3, so we
have negative 1 over 3 e raised to the power of 3x. Now I will evaluate this at
x equals 1 and then 0 and then find the difference. So when x equals 1,
we have 1 over 4 e raised to the power of 4 minus 1 over 3 e raised to the power of 3.
Then when x is 0, we have 1 over 4 e raised to the power of 0 minus
1 over 3 e raised to the power of 0. So this would give us 1 over 4 e raised to the power of 4
minus 1 over 3 e raised to the power of 3. Then here we'll have minus 1 over 4 and
then plus 1 over 3 since e raised to the power of 0 is equal to 1.
Now I combine these fractions. Our common denominator would be 12. Multiply this by
3 over 3 and multiply this by 4 over 4. So we have 1 over 4 e raised to the power of 4 minus
1 over 3 e raised to the power of 3. This would be minus 3 over 12 plus 4 over 12, so this would
be the exact area. Let's also get a decimal approximation. This is approximately 7.3769.
Now in the homework, though, we should enter the exact value unless it tells us to round.
So going back to our graph one more time, the area of the shaded region is exactly
1 over 4 e raised to the power of 4 minus 1 over 3 e raised to the power of 3 plus 1 over 12, which
means the area is approximately 7.3769 square units.
I hope you found this helpful.