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4. (a) Express $$ ext{lim}_{eta o 0} rac{2}{eta} extstyle extstyle extsum_{x=2.1}^{6.3} ext{dx}$$ as an integral - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 1
Question 6
4. (a) Express $$ ext{lim}_{eta o 0} rac{2}{eta} extstyle extstyle extsum_{x=2.1}^{6.3} ext{dx}$$ as an integral. (b) Hence show that $$ ext{lim}_{eta o... show full transcript
Worked Solution & Example Answer:4. (a) Express $$ ext{lim}_{eta o 0} rac{2}{eta} extstyle extstyle extsum_{x=2.1}^{6.3} ext{dx}$$ as an integral - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 1
Step 1
Express $$ ext{lim}_{eta o 0} rac{2}{eta} extstyle extsum_{x=2.1}^{6.3} ext{dx}$$ as an integral.
Answer
To express the limit as an integral, we recognize that the summation can be approximated as follows:
We can interpret the summation as a Riemann sum: ext{lim}_{eta o 0} rac{2}{eta} extstyle extsum_{x=2.1}^{6.3} ext{dx} = ext{lim}_{n o ext{∞}} extstyle extsum_{i=1}^{n} f(x_i) riangle x
Here, we define:
- (the value being summed)
- riangle x = rac{3.2}{n} (the width of each subinterval)
Thus, the above summation can be expressed as: ext{lim}_{n o ext{∞}} extsum_{i=1}^{n} f(x_i) riangle x = ext{lim}_{eta o 0} rac{2}{eta} extstyle extsum_{x=2.1}^{6.3} ext{dx} = extstyle extint_{2.1}^{6.3} 2 ext{dx}
where the limits 2.1 and 6.3 represent the bounds of integration.
Step 2
Hence show that $$ ext{lim}_{eta o 0} rac{2}{eta} extstyle extsum_{x=2.1}^{6.3} ext{dx} = ext{ln} k$$ where k is a constant to be found.
Answer
Starting from the integral we just found:
Evaluating this definite integral:
We can express this result in terms of logarithms:
Since we need to demonstrate that the limit equals to ln(k), we consider:
Therefore, we find: .
Thus, concluding that: ext{lim}_{eta o 0} rac{2}{eta} extstyle extsum_{x=2.1}^{6.3} ext{dx} ext{ equals } ext{ln}(e^{8.4}) ext{, confirming } k = e^{8.4}.