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Figure 1 shows part of the curve with equation $y = e^{0.5x}$ - Edexcel - A-Level Maths: Pure - Question 4 - 2008 - Paper 7
Question 4
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 4 - 2008 - Paper 7
Step 1
Complete the table with the values of $y$ corresponding to $x = 0.8$ and $x = 1.6$
Answer
To find the values for when and , we can use the equation :
For : (rounded to 5 significant figures)
For : (rounded to 5 significant figures)
Thus, the completed table with approximate values is:
| 0 | 0.4 | 0.8 | 1.2 | 1.6 | 2 | |
|---|---|---|---|---|---|---|
Step 2
Use the trapezium rule with all the values in the table to find an approximate value for the area of $R$
Answer
To apply the trapezium rule, we need to calculate the area using the formula: A = rac{1}{2}ht(a+b) + h \\sum_{i=1}^{n-1}y_i where is the area, is the width of the trapezium, and are the heights of the first and last sections respectively, and are the intermediate heights.
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Width of each sub-interval (h):
- The interval is , thus .
- The values of from the table are approximately:
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Using the trapezium rule:
- Area: A = rac{1}{2} (0.4) (1.0 + 7.38906) + 0.4 (1.49182 + 1.64872 + 2.22554 + 3.68078)
- Calculating the sums:
Thus, the approximate value for the area of is: