Video: Expressing the Limit of a Riemann Sum in the Notation of Definite Integration

Express lim_(𝑛 β†’ ∞) βˆ‘_(𝑖 = 1)^(𝑛) 𝑒^(π‘₯𝑖)/(2 βˆ’ 4π‘₯_(𝑖)) Ξ”π‘₯_(𝑖) as a definite integral on the closed interval [βˆ’5, βˆ’3].

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Video Transcript

Express the limit as 𝑛 approaches ∞ of the sum of 𝑒 to the power of π‘₯𝑖 over two minus four π‘₯𝑖 times Ξ”π‘₯𝑖 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 π‘Ž to 𝑏, then the definite integral between 𝑏 and π‘Ž of 𝑓 of π‘₯ with respect to π‘₯ is equal to the limit as 𝑛 approaches ∞ of the sum of 𝑓 of π‘₯𝑖 times Ξ”π‘₯ 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 π‘Ž 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 π‘₯𝑖 is equal to 𝑒 to the power of π‘₯𝑖 over two minus four π‘₯𝑖. Well, that’s great because that means 𝑓 of π‘₯ is equal to 𝑒 to the power of π‘₯ over two minus four π‘₯. 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 π‘₯ over two minus four π‘₯ with respect to π‘₯.

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