Figure 1 shows part of the curve with equation $y = e^{0.5x}$. The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the x-axis, the y-axis and the line $x = 2$. (a) Complete the table with the values of $y$ corresponding to $x = 0.8$ and $x = 1.6$. | $x$ | $y$ | |------|-----| | 0 | $e^0$ | | 0.4 | $e^{0.4}$ | | 0.8 | $e^{0.8}$ | | 1.2 | $e^{1.2}$ | | 1.6 | $e^{1.6}$ | | 2 | $e^2$ | (b) Use the trapezium rule with all the values in the table to find an approximate value for the area of $R$, giving your answer to 4 significant figures.

A-Level Edexcel Maths Pure Differentiation: Figure 1 shows part of the curve

Figure 1 shows part of the curve with equation $y = e^{0.5x}$ - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 7

Question 3

$Figure-1-shows-part-of-the-curve-with-equation-$y-=-e^{0.5x}$-Edexcel-A-Level Maths Pure-Question 3-2008-Paper 7.png$

Figure 1 shows part of the curve with equation $y = e^{0.5x}$. The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the x-axis, the y-axis and t... show full transcript

Worked Solution & Example Answer:Figure 1 shows part of the curve with equation $y = e^{0.5x}$ - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 7

Step 1

Complete the table with the values of $y$ corresponding to $x = 0.8$ and $x = 1.6$

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Answer

$x$	$y$
0	$1$
0.4	$1.4918$
0.8	$2.2255$
1.2	$3.3201$
1.6	$4.9530$
2	$7.3891$

Step 2

Use the trapezium rule to find an approximate value for the area of $R$

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Answer

To apply the trapezium rule:

Identify the values of $y$ : $y_0 = 1$ , $y_1 = 1.4918$ , $y_2 = 2.2255$ , $y_3 = 3.3201$ , $y_4 = 4.9530$ , $y_5 = 7.3891$ .
The area is calculated as:

$ext{Area} = rac{1}{2} imes ext{interval width} imes (y_0 + 2y_1 + 2y_2 + 2y_3 + 2y_4 + y_5)$
The interval width is $0.4$ ;

$ext{Area} = rac{1}{2} imes 0.4 imes (1 + 2(1.4918) + 2(2.2255) + 2(3.3201) + 2(4.9530) + 7.3891)$
Performing the calculations:

$ext{Area} = 0.2 imes (1 + 2.9836 + 4.4510 + 6.6402 + 9.9060 + 7.3891) = 0.2 imes 32.3709 = 4.4744$
Rounding to 4 significant figures gives:

$ext{Area} ext{ of } R ext{ is approximately } 4.474$ .

Figure 1 shows part of the curve with equation $y = e^{0.5x}$ - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 7

Worked Solution & Example Answer:Figure 1 shows part of the curve with equation $y = e^{0.5x}$ - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 7

Complete the table with the values of $y$ corresponding to $x = 0.8$ and $x = 1.6$

Use the trapezium rule to find an approximate value for the area of $R$

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