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Figure 1 shows part of the curve with equation $y = ext{(0.75 + cos } x)$ - Edexcel - A-Level Maths Pure - Question 3 - 2010 - Paper 6
Question 3
Figure 1 shows part of the curve with equation $y = ext{(0.75 + cos } x)$. The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the y-axis, the... show full transcript
Worked Solution & Example Answer:Figure 1 shows part of the curve with equation $y = ext{(0.75 + cos } x)$ - Edexcel - A-Level Maths Pure - Question 3 - 2010 - Paper 6
Step 1
Complete the table with values of $y$ corresponding to $x = rac{ ext{π}}{6}$ and $x = rac{ ext{π}}{4}$.
Answer
To find the values of , we will evaluate the curve equation at the specified values:
For x = rac{ ext{π}}{6}:
\approx 1.2247$$ For $x = rac{ ext{π}}{4}$: $$y = 0.75 + ext{cos} rac{ ext{π}}{4} = 0.75 + rac{ ext{1}}{ ext{√2}} \\ \approx 1.000$$ Thus, the completed table is: | $x$ | $0$ | $ rac{ ext{π}}{12}$ | $ rac{ ext{π}}{6}$ | $ rac{ ext{π}}{4}$ | $ rac{ ext{π}}{3}$ | |------------------|----------------|------------------------|----------------------|---------------------|-------------------| | $y$ | $1.3229$ | $1.2973$ | $1.2247$ | $1.000$ | $1.180$ |Step 2
Use the trapezium rule (i)
Answer
To estimate the area of using the trapezium rule for , x = rac{ ext{π}}{6}, and x = rac{ ext{π}}{3}:
We apply the formula: I = rac{ ext{h}}{2} (f(a) + f(b)) where , b = rac{ ext{π}}{6} and h = rac{ ext{π}}{6}.
Calculating:
\ \ \ \ \ \ = rac{ ext{π}}{12} imes (1.3229 + 1.180) \\ \ \ \ \ \ = rac{ ext{π}}{12} imes 2.5029 \\ \ = rac{2.5029 imes ext{π}}{12} \\ \ \approx 1.249$$Step 3
Use the trapezium rule (ii)
Answer
To further estimate the area of with values at , x = rac{ ext{π}}{12}, x = rac{ ext{π}}{6}, x = rac{ ext{π}}{4} and x = rac{ ext{π}}{3}:
This time, the interval lengths will be different, with sub-intervals:
- I_1 = rac{ ext{h}}{2} (f(x_0) + f(x_1))
- I_2 = rac{ ext{h}}{2} (f(x_1) + f(x_2))
- I_3 = rac{ ext{h}}{2} (f(x_2) + f(x_3))
- I_4 = rac{ ext{h}}{2} (f(x_3) + f(x_4))
Thus, we calculate:
\approx 1.257$$