# Question Video: Expressing the Limit of a Riemann Sum in the Notation of Definite Integration Mathematics • Higher Education

Express lim_(𝑛 → ∞) ∑_(𝑖 = 1)^(𝑛) 𝑒^(𝑥𝑖)/(2 − 4𝑥_(𝑖)) Δ𝑥_(𝑖) as a definite integral on the closed interval [−5, −3].

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### Video Transcript

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.