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The table below shows corresponding values of x and y for y = log_2 x - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Question 6

The table below shows corresponding values of x and y for y = log_2 x. The values of y are given to 2 decimal places as appropriate. (a) Using the trapezium rule w... show full transcript

Worked Solution & Example Answer:The table below shows corresponding values of x and y for y = log_2 x - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Step 1

Using the trapezium rule with all the values of y in the table, find an estimate for ∫_3^8 log_2 2x dx

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Answer

The trapezium rule can be stated as:

$T = \frac{h}{2} \left( y_0 + 2y_1 + 2y_2 + ... + 2y_{n-1} + y_n \right)$

where h is the width of each sub-interval. Here, the values are given for x from 3 to 8, so:

The intervals are 3, 4.5, 6, 7.5, and 8, giving:
- h = 8 - 3 = 5
- There are 4 intervals: 3 to 4.5, 4.5 to 6, 6 to 7.5, and 7.5 to 8.

We need to calculate h = 1.5 and then apply the trapezium rule:

$T = \frac{1.5}{2} \left( y_0 + 2y_1 + 2y_2 + 2y_3 + y_4 \right)$

Substituting the corresponding values of y from the table:

y_0 = 1.63 (for x=3)
y_1 = 2.26 (for x=4.5)
y_2 = 2.46 (for x=6)
y_3 = 2.63 (for x=7.5)
y_4 = 2.63 (for x=8)

Thus, the estimate becomes:

$T = \frac{1.5}{2} \left( 1.63 + 2(2.26) + 2(2.46) + 2.63 \right)$

Calculating this gives:

$T = \frac{1.5}{2} \left( 1.63 + 4.52 + 4.92 + 2.63 \right) = \frac{1.5}{2} \cdot 13.8 = 10.35$

Therefore, the estimate for the integral is approximately 10.35.

Step 2

Using your answer to part (a) and making your method clear, estimate (i) ∫_3^8 log_2 (2x)^{10} dx

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Answer

To estimate ∫_3^8 log_2 (2x)^{10} dx, we first rewrite the integrand:

$\log_2 ((2x)^{10}) = 10 \log_2 (2x)$

Substituting into the integral gives:

$10 \int_3^8 \log_2 (2x) dx$

Using the result from part (a), we have:

$\int_3^8 \log_2 (2x) dx = 10.35$

Thus,

$10 \int_3^8 \log_2 (2x) dx \approx 10 \cdot 10.35 = 103.5$

Therefore, the estimate for the integral is approximately 103.5.

Step 3

Using your answer to part (a) and making your method clear, estimate (ii) ∫_3^8 log_2 18x dx

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Answer

For the integral ∫_3^8 log_2 18x dx, we can separate the logarithm:

$\log_2 (18x) = \log_2 18 + \log_2 x$

Thus, the integral can be split:

$\int_3^8 \log_2 (18x) dx = \int_3^8 \log_2 18 \, dx + \int_3^8 \log_2 x \, dx$

Calculating the first integral:

$\int_3^8 \log_2 18 \, dx = \log_2 18 (8 - 3) = \log_2 18 \cdot 5 = 5 \log_2 (18)$

For the second integral, we use the trapezium rule as in part (a). We substitute the values based on the table:

$\int_3^8 \log_2 x \approx 10.35$

Therefore, the combined estimate is:

$5 \log_2 (18) + 10.35$

If we calculate \log_2 (18) (approximately 4.17) and compute:

$5 \cdot 4.17 + 10.35 = 20.85 + 10.35 = 31.2$

Thus, the estimate for the second integral is approximately 31.2.

The table below shows corresponding values of x and y for y = log_2 x - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Worked Solution & Example Answer:The table below shows corresponding values of x and y for y = log_2 x - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Using the trapezium rule with all the values of y in the table, find an estimate for ∫_3^8 log_2 2x dx

Using your answer to part (a) and making your method clear, estimate (i) ∫_3^8 log_2 (2x)^{10} dx

Using your answer to part (a) and making your method clear, estimate (ii) ∫_3^8 log_2 18x dx

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