A-level · Integration

Areas with integrationExam-style questions

Setting up a definite integral for each kind of area question. A region between a curve and the x-axis, with the limits found from the x-intercepts; a region below the axis, where the integral is negative and the area is its size; a curve that crosses the axis inside the range, split at the crossing; the region between a curve and a line, as top minus bottom between the points where they meet; and regions made of a triangle and a curved piece.

Start practising Read the lesson

29 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
Figure 1Figure 1 shows a sketch of part of the curve $C$ with equation

$y = x^{\frac{3}{2}} - 3x \qquad x \geqslant 0$

The curve meets the $x$ -axis at the origin $O$ and at the point $A$ , and has a minimum point at $B$ .
(a) Use calculus to find the $x$ coordinate of $B$ .

(4 marks)

In Figure 1, the shaded region $R_1$ lies between $C$ and the $x$ -axis from $O$ to $A$ .

The shaded region $R_2$ lies between $C$ and the $x$ -axis from $A$ to the line with equation $x = k$ , where $k$ is a constant.

Given that $R_1$ and $R_2$ have the same area,
(b) use algebraic integration to find the exact value of $k$ .

(4 marks)
Level 4 · 8 marksTry it
2
The curve $C$ has equation $y = x^3 - 6x^2 + 9x$ .(a) Show that $y = x(x - 3)^2$ , and hence write down the coordinates of the points where $C$ meets the $x$ -axis. (2 marks)
(b) Find the coordinates of the stationary points of $C$ , and determine the nature of each. (4 marks)
(c) Find the area of the finite region enclosed by $C$ and the $x$ -axis. (3 marks)
Level 4 · 9 marksTry it
3
Figure 2

Figure 2 shows the curve $y = 16 - x^{2}$ and the line $y = 6x$ . The line meets the curve at the point $A$ in the first quadrant, and the curve crosses the positive $x$ -axis at $B$ .
(a) Find the coordinates of $A$ and of $B$ . (3 marks)
(b) Find the area of the region bounded by the line $OA$ , the arc of the curve from $A$ to $B$ , and the $x$ -axis. (5 marks)
Level 4 · 8 marksTry it
4
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

$y = \frac{(x - 1)(6 - x)}{\sqrt{x}} \qquad x > 0$

The curve crosses the $x$ -axis at the points where $x = 1$ and $x = 6$

The finite region $R$ is bounded by the curve and the $x$ -axis.

Find the exact area of $R$ , writing your answer in the form $a\sqrt{6} + b$ , where $a$ and $b$ are constants to be found.

(6 marks)
Level 4 · 6 marksTry it
5
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

$y = \frac{6}{x} \qquad x > 0$

The line $l$ has equation $x + y = 7$

The line $l$ meets the curve $C$ at the points $A$ and $B$ .
(a) Find the coordinates of $A$ and $B$ .

(3 marks)

The finite region $R$ is bounded by the line $l$ and the curve $C$ .
(b) Use algebraic integration to find the exact area of $R$ , giving your answer in the form $a + b\ln 6$ , where $a$ and $b$ are rational constants to be found.

(4 marks)
Level 4 · 7 marksTry it

2 more questions on areas with integration

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free