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 Ξπ₯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 Ξπ₯ 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.
Letβs 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, thatβs 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. Itβs 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₯.
