Video Transcript
Estimate the definite integral
between the limits of one and two of 𝑒 to the power of 𝑥 over 𝑥 d𝑥, using the
trapezoidal rule with four subintervals. Approximate your answer to two
decimal places.
Remember, the trapezoidal rule says
that we can find an estimate for the definite integral of some function 𝑓 of 𝑥
between the limits of 𝑎 and 𝑏 by performing the calculation Δ𝑥 over two times 𝑓
of 𝑥 nought plus 𝑓 of 𝑥 𝑛 plus two times 𝑓 of 𝑥 one plus 𝑓 of 𝑥 two all the
way through to 𝑓 of 𝑥 𝑛 minus one. Where Δ𝑥 is 𝑏 minus 𝑎 over 𝑛,
where 𝑛 is the number of subintervals. And 𝑥 𝑖 is 𝑎 plus 𝑖 lots of
Δ𝑥. We’ll begin then just simply by
working out Δ𝑥. Contextually, Δ𝑥 is the width of
each of our subinterval. Here we’re working with four
subintervals. So 𝑛 is equal to four. 𝑎 is equal to one. And 𝑏 is equal to two. Δ𝑥 is therefore two minus one over
four, which is a quarter or 0.25. That’s the perpendicular height of
each trapezoid.
The values for 𝑓 of 𝑥 nought and
𝑓 of 𝑥 one and so on require a little more work. But we can make this as simple as
possible by including a table. We recall that there will always be
one more 𝑓 of 𝑥 value than the number of subintervals. So here, that’s going to be four
plus one, which is five 𝑓 of 𝑥 values. The 𝑥-values themselves run from
𝑎 to 𝑏. That’s here from one to two. And the ones in between are found
by repeatedly adding Δ𝑥, that’s 0.25, to 𝑎, which is one. So these values are 1.25, 1.5, and
1.75. And that gives us our four strips
of width 0.25 units. We’re then going to substitute each
𝑥-value into our function.
Here, we’re going to need to make a
decision on accuracy. Whilst the question tells us to use
an accuracy of two decimal places, that’s only for our answer. A good rule of thumb is to use at
least five decimal places. We begin with 𝑓 of one. That’s 𝑒 to the power of one over
one, which is 2.71828, correct to five decimal places. We have 𝑓 of 1.25, which is 𝑒 to
the power of 1.25 divided by 1.25. That’s, correct to five decimal
places, 2.79227. We repeat this process for 1.5. 𝑓 of 1.5 is 2.98779. 𝑓 of 1.75 is 3.28834. And 𝑓 of two is 3.69453 rounded to
five decimal places. All that’s left is to substitute
what we know into our formula for the trapezoidal rule. It’s Δ𝑥 over two. That’s 0.25 over two times 𝑓 of
one. That’s 2.71828 plus 𝑓 of two. That’s 3.69453 plus two lots of
everything else essentially. That’s 2.79227, 2.98779, and
3.28834. That gives us 3.0687, which,
correct to two decimal places, is 3.07.
It’s useful to remember that we can
check whether this answer is likely to be sensible by using the integration function
on our calculator. And when we do, we get 3.06 correct
to two decimal places. That’s really close to the answer
we got, suggesting we’ve probably performed our calculations correctly. And so an approximation to the
integral evaluated between one and two of 𝑒 to the power of 𝑥 over 𝑥 d𝑥 is
3.07.
