Express the limit as 𝑛 approaches
∞ of the sum of 𝑒 to the power of 𝑥𝑖 over two minus four 𝑥𝑖 times Δ𝑥𝑖
for values of 𝑖 from one to 𝑛 as a definite integral on the closed interval
negative five to negative three.
Remember, if 𝑓 is integrable on
some closed interval 𝑎 to 𝑏, then the definite integral between 𝑏 and 𝑎 of 𝑓 of
𝑥 with respect to 𝑥 is equal to the limit as 𝑛 approaches ∞ of the sum of
𝑓 of 𝑥𝑖 times Δ𝑥 for values of 𝑖 from one to 𝑛. Now we can quite clearly see that
our interval is from negative five to negative three inclusive. So we begin by letting 𝑎 be equal
to negative five and 𝑏 be equal to negative three.
Let’s now compare our limit to the
general form. We can see that 𝑓 of 𝑥𝑖 is equal
to 𝑒 to the power of 𝑥𝑖 over two minus four 𝑥𝑖. Well, that’s great because that
means 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥 over two minus four 𝑥. This means the limit of our Riemann
sums can be expressed as a definite integral. It’s the definite integral between
negative five and negative three of 𝑒 to the power of 𝑥 over two minus four 𝑥
with respect to 𝑥.