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

Express lim_(π β β) β_(π = 1)^(π) π^(π₯π)/(2 β 4π₯_(π)) Ξπ₯_(π) as a definite integral on the closed interval [β5, β3].

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Video Transcript

Express the limit as π approaches β of the sum of π to the power of π₯π over two minus four π₯π times Ξπ₯π for values of π from one to π as a definite integral on the closed interval negative five to negative three.

Remember, if π is integrable on some closed interval π to π, then the definite integral between π and π of π of π₯ with respect to π₯ is equal to the limit as π approaches β of the sum of π of π₯π times Ξπ₯ for values of π from one to π. Now we can quite clearly see that our interval is from negative five to negative three inclusive. So we begin by letting π be equal to negative five and π be equal to negative three.

Letβs now compare our limit to the general form. We can see that π of π₯π is equal to π to the power of π₯π over two minus four π₯π. Well, thatβs great because that means π of π₯ is equal to π 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. Itβs the definite integral between negative five and negative three of π to the power of π₯ over two minus four π₯ with respect to π₯.