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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
Step 2
Use the trapezium rule (i) with the values of $y$ at $x = 0$, $x = rac{ ext{π}}{6}$ and $x = rac{ ext{π}}{3}$
Answer
The trapezium rule formula is given by:
I ext{ }= ext{ } rac{b-a}{2} imes (f(a) + f(b))
For this case, we have:
- , b = rac{ ext{π}}{6}
- , figg( rac{ ext{π}}{6}igg) = 1.2973, figg( rac{ ext{π}}{3}igg) = 1.180
Thus, the estimate of the area of is approximately
rac{ rac{ ext{π}}{6} - 0}{2} imes (1.3229 + 1.2973 + 1.180) ext{ } \\ ext{ } = rac{ ext{π}}{12} imes (1.3229 + 1.2973 + 1.180) ext{ } \\ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \\ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \\ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \text{ } \\ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \text{ } \\ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \text{ } \\ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \text{ } \\ ext{ } \text{ } \ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \text{ } \text{ } \ ext{ } \text{ } 1.249 ext{ (to 3 decimal places)}
Step 3
Use the trapezium rule (ii) with the values of $y$ at $x = 0$, $x = rac{ ext{π}}{12}$, $x = rac{ ext{π}}{6}$, $x = rac{ ext{π}}{4}$ and $x = rac{ ext{π}}{3}$
Answer
Using a more detailed trapezium rule:
I ext{ }= ext{ } rac{b-a}{n} imes igg( f(a) + 2igg(figg( rac{ ext{π}}{12}igg) + figg( rac{ ext{π}}{6}igg) + figg( rac{ ext{π}}{4}igg)igg) + f(b) igg)
Where:
- , b = rac{ ext{π}}{3}
- , figg( rac{ ext{π}}{12}igg) = 1.2247, figg( rac{ ext{π}}{6}igg) = 1.2973, figg( rac{ ext{π}}{4}igg) = 1.2247, figg( rac{ ext{π}}{3}igg) = 1.180
Thus, we can calculate:
I = rac{ rac{ ext{π}}{3}}{4} imes igg( 1.3229 + 2(1.2247 + 1.2973 + 1.2247) + 1.180 igg) \ ext{ } ext{ } \ ext{} ext{ } \ ext{} \ ext{} \ ext{} \ \text{ } ext{ } 1.257 ext{ (to 3 decimal places)}