Question 4 - A-Level Maths

Edexcel C4 — Question 4 9 marks

Exam Board	Edexcel
Module	C4 (Core Mathematics 4)
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Indefinite & Definite Integrals
Type	Numerical integration comparison
Difficulty	Standard +0.3 This is a straightforward application of the trapezium rule with standard intervals (2 then 4), requiring calculator work and function evaluation. Part (c) adds mild conceptual understanding about concavity, but the overall question is slightly easier than average as it's purely procedural with no algebraic manipulation or problem-solving insight required.
Spec	1.09f Trapezium rule: numerical integration

4. (a) Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral $$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$ giving your answer to 3 significant figures.
(b) Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.
(c) Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve $y = \mathrm { e } ^ { \cos x }$ in the interval $0 \leq x \leq 3$ and explain your answer.
4. continued

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Answer	Marks	Guidance
$x$	$0$	$0.75$
$y$	$2.7183$	$2.0786$
B2
(a) $= \frac{1}{2} \times 1.5 \times [2.7183 + 0.3716 + 2(1.0733)] = 3.93$ (3sf)	B1 M1 A1

(b) $= \frac{1}{2} \times 0.75 \times [2.7183 + 0.3716 + 2(2.0786 + 1.0733 + 0.5336)]$

Answer	Marks	Guidance
$= 3.92$ (3sf)	M1 A1
(c) Curve must be above top of trapezia in some places and below in others hence position of ordinates determines whether estimate is high or low	B2	(9 marks)

| $x$ | $0$ | $0.75$ | $1.5$ | $2.25$ | $3$ |
|-----|-----|--------|-------|--------|------|
| $y$ | $2.7183$ | $2.0786$ | $1.0733$ | $0.5336$ | $0.3716$ |

| B2 |

**(a)** $= \frac{1}{2} \times 1.5 \times [2.7183 + 0.3716 + 2(1.0733)] = 3.93$ (3sf) | B1 M1 A1 |

**(b)** $= \frac{1}{2} \times 0.75 \times [2.7183 + 0.3716 + 2(2.0786 + 1.0733 + 0.5336)]$
$= 3.92$ (3sf) | M1 A1 |

**(c)** Curve must be above top of trapezia in some places and below in others hence position of ordinates determines whether estimate is high or low | B2 | (9 marks)

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4. (a) Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral

$$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$

giving your answer to 3 significant figures.\\
(b) Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.\\
(c) Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve $y = \mathrm { e } ^ { \cos x }$ in the interval $0 \leq x \leq 3$ and explain your answer.\\

4. continued\\

\hfill \mbox{\textit{Edexcel C4  Q4 [9]}}

This paper (8 questions)

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Q1 6 Q2 7 Q3 8 Q4 9 Q5 9 Q6 10 Q7 11 Q8 15

Answer	Marks	Guidance
\(x\)	\(0\)	\(0.75\)
\(y\)	\(2.7183\)	\(2.0786\)
B2
(a) \(= \frac{1}{2} \times 1.5 \times [2.7183 + 0.3716 + 2(1.0733)] = 3.93\) (3sf)	B1 M1 A1

Answer	Marks	Guidance
\(= 3.92\) (3sf)	M1 A1
(c) Curve must be above top of trapezia in some places and below in others hence position of ordinates determines whether estimate is high or low	B2	(9 marks)

Edexcel C4 — Question 4 9 marks

(b) \(= \frac{1}{2} \times 0.75 \times [2.7183 + 0.3716 + 2(2.0786 + 1.0733 + 0.5336)]\)

This paper (8 questions)