A-level · Integration

Definite integration and area under a curveExam-style questions

Finding the area of a region with a curved edge. Thin strips and the limit of a sum; why an antiderivative measures area (the fundamental theorem); evaluating with square brackets, top limit minus bottom limit, with no + C; why a region below the x-axis counts as negative, so an integral and an area can differ; splitting and reversing the limits; and finding a limit that is a letter.

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5 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
(a) Express

$\lim_{\delta x \to 0} \sum_{x = 3}^{12} \frac{5}{x}\,\delta x$

as an integral.

(1 mark)
(b) Hence show that

$\lim_{\delta x \to 0} \sum_{x = 3}^{12} \frac{5}{x}\,\delta x = \ln k$

where $k$ is a constant to be found.

(2 marks)
Level 2 · 3 marksTry it
2
The curve $y = x(4 - x)$ and the line $y = x$ enclose a finite region.
(a) Find the $x$ -coordinates of the points where they meet.
(b) Find the area of the region enclosed between them.
Level 3 · 5 marksTry it
3
The region $R$ is bounded by the curve $y = \dfrac{4}{1 + x^4}$ , the $x$ -axis, the $y$ -axis and the line $x = 1$ . Priya estimates the area of $R$ by splitting the interval $0 \le x \le 1$ into strips of width $\delta x = 0.2$ . On each strip she draws two rectangles, one lying entirely below the curve and one whose top lies entirely above the curve, as shown in the diagram.(a) Show that Priya's rectangles give a lower bound of $3.25$ and an upper bound of $3.65$ for the area of $R$ , correct to 3 significant figures.

(1 mark)
(b) Priya repeats the process, halving the width of the strips each time. Her lower and upper bounds, correct to 3 significant figures, are given in the table.
Width of strip, $\delta x$ $0.2$ $0.1$ $0.05$ $0.025$
Lower bound $3.25$ $3.36$ $3.42$ $3.44$
Upper bound $3.65$ $3.56$ $3.52$ $3.49$
Use the table to estimate the area of $R$ correct to 2 significant figures, giving a reason for your estimate.

(2 marks)
(c) Write down an expression in $y$ and $\delta x$ that gives the area of $R$ exactly.

(2 marks)
Level 3 · 5 marksTry it
4
(a) Ravi and Ellen were each asked to find $\displaystyle\int (3x - 2)^2\,dx$ .

Ravi wrote: $\displaystyle\int (3x - 2)^2\,dx = \tfrac19(3x - 2)^3 + c$

Ellen wrote: $\displaystyle\int (3x - 2)^2\,dx = \int (9x^2 - 12x + 4)\,dx = 3x^3 - 6x^2 + 4x + c$

Ravi multiplies out his answer to get $3x^3 - 6x^2 + 4x - \tfrac89 + c$ . He says that Ellen's answer must be wrong because it does not contain the term $-\tfrac89$ .

Explain whether you agree with Ravi.

(1 mark)
(b) In this part of the question $k$ is a constant greater than $1$ .

(i) Find $\displaystyle\int_1^k \frac{12}{(2x + 1)^2}\,dx$ . Express your answer as a single fraction in terms of $k$ .

(ii) The region bounded by the curve $y = \dfrac{12}{(2x + 1)^2}$ , the $x$ -axis and the lines $x = 1$ and $x = k$ has area $\dfrac43$ .

Determine the value of $k$ .

(5 marks)
(c) In this question you must show detailed reasoning.

Find, in its simplest form, the exact value of $\displaystyle\int_0^{\frac{\pi}{4}} \frac{\sec^2 x}{1 + 3\tan x}\,dx$ .

(4 marks)
Level 4 · 10 marksTry it
5
The diagram shows part of the curve $y = e^{-\sqrt{x}}$ .(a) Use rectangles of width $0.75$ to find upper and lower bounds for $\displaystyle\int_1^4 e^{-\sqrt{x}}\,dx$ . Give your answers correct to 3 significant figures.

(4 marks)
(b) (i) Use the substitution $t = \sqrt{x}$ to show that $\displaystyle\int_1^4 e^{-\sqrt{x}}\,dx = \displaystyle\int_1^2 2te^{-t}\,dt$ .

(ii) Hence show that $\displaystyle\int_1^4 e^{-\sqrt{x}}\,dx = 4e^{-1} - 6e^{-2}$ .

(7 marks)
Level 5 · 11 marksTry it

Width of strip, $\delta x$	$0.2$	$0.1$	$0.05$	$0.025$
Lower bound	$3.25$	$3.36$	$3.42$	$3.44$
Upper bound	$3.65$	$3.56$	$3.52$	$3.49$