Complete the table below, giving the values of y to 2 decimal places.

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Complete the table below, giving the values of y to 2 decimal places - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 3

Question 8

Complete the table below, giving the values of y to 2 decimal places. | x | 2 | 2.25 | 2.5 | 2.75 | 3 | |-------|---|------|-----|------|---| | y | 0.5 | 0.... show full transcript

Worked Solution & Example Answer:Complete the table below, giving the values of y to 2 decimal places - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 3

Step 1

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Answer

To complete the table, we need to calculate the values of $y$ using the formula $y = \frac{5}{3x^{2}-2}$ for each corresponding $x$ value:

For $x = 2$ : $y = \frac{5}{3(2^{2})-2} = \frac{5}{10} = 0.5$
For $x = 2.25$ : $y = \frac{5}{3(2.25^{2})-2} = \frac{5}{15.1875 - 2} \approx 0.38$
For $x = 2.5$ : $y = \frac{5}{3(2.5^{2})-2} = \frac{5}{18.75 - 2} \approx 0.32$
For $x = 2.75$ : $y = \frac{5}{3(2.75^{2})-2} = \frac{5}{22.6875 - 2} \approx 0.24$
For $x = 3$ : $y = \frac{5}{3(3^{2})-2} = \frac{5}{27 - 2} = 0.2$

The completed table is:

x	2	2.25	2.5	2.75	3
y	0.5	0.38	0.32	0.24	0.2

Step 2

Use the trapezium rule, with all the values of y from your table, to find an approximate value for $\int_{2}^{3} \frac{5}{3x^{2}-2} \,dx$.

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Answer

The trapezium rule formula is given by: $A \approx \frac{b-a}{2}(f(a) + f(b))$ where $f(a)$ and $f(b)$ are the function values at $a$ and $b$ , respectively. For this question, we will also account for the other values:

The interval $[2, 3]$ consists of values at $x = 2, 2.25, 2.5, 2.75, 3$ . We compute the area:

$h = 0.25$ (the width of each sub-interval)
$y$ values are: $f(2)=0.5$ , $f(2.25)=0.38$ , $f(2.5)=0.32$ , $f(2.75)=0.24$ , and $f(3)=0.2$

Calculating the total area: $A \approx \frac{0.25}{2}(0.5 + 0.38 + 0.32 + 0.24 + 0.2) = 0.125 (0.5 + 0.38 + 0.32 + 0.24 + 0.2) = 0.125 \times 1.64 = 0.205$

Thus, the approximate value for the integral is $0.205$ .

Step 3

Use your answer to part (b) to find an approximate value for the area of S.

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Answer

To find the area of region S, we have the trapezium rule calculated area as approximately $0.205$ from part (b). Since the area S is entirely below the curve and above the x-axis and the line through B at (2, 0), we can directly state that the approximate value for the area of S is $0.205$ .

Complete the table below, giving the values of y to 2 decimal places - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 3

Worked Solution & Example Answer:Complete the table below, giving the values of y to 2 decimal places - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 3