A-Level Edexcel Maths Pure Applications of 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

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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

To find the values of $y$ for the given $x$ values:

For $x = 0.8$ , we calculate:

$y = e^{0.5 imes 0.8} = e^{0.4} \approx 1.4918$ (rounded to 4 significant figures)
For $x = 1.6$ , we calculate:

$y = e^{0.5 imes 1.6} = e^{0.8} \approx 2.2255$ (rounded to 4 significant figures)

Thus, the completed table will look like:

$x$	$0$	$0.4$	$0.8$	$1.2$	$1.6$	$2$
$y$	$e^{0}$	$e^{0.4}$	$1.4918$	$?$	$2.2255$	$e^{2}$

Step 2

Use the trapezium rule with all the values in the table to find an approximate value for the area of $R$

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Answer

To calculate the area under the curve using the trapezium rule, we will use the formula:

$ext{Area} = \frac{1}{2} (b_1 + b_n) + \sum_{i=1}^{n-1} b_i$

where $b_1$ and $b_n$ are the first and last ordinates and $b_i$ are the intermediate ordinates. The required values are:

First ordinate ( $x=0$ ): $y = e^{0} = 1$
Last ordinate ( $x=2$ ): $y = e^{2} \approx 7.3891$
Intermediate ordinates: $e^{0.4} \approx 1.4918$ , $1.4918$ , $2.2255$

Now we calculate the area:

$\text{Area} = \frac{1}{2}(1 + 7.3891) + (1.4918 + 1.4918 + 2.2255)$

Calculate: $\text{Area} \approx 0.2 \times (1 + 2.2255 + 7.3891)$

Thus, summing gives:

$\text{Area} \approx 4.922206$

Rounding to 4 significant figures, the approximate area is $4.922$ .

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 with all the values in the table to find an approximate value for the area of $R$

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