CIE Area between curves (-2020)

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1. $(\mathrm{CIF} 0606 / 2018 / \mathrm{w} / 13 / \mathrm{q} 10)$

The diagram shows the curve $y=12+x-x^{2}$ intersecting the line $y=x+8$ at the points $A$ and $B$.

(i) Find the coordinates of the points $A$ and $B$. $[3]$

(ii) Find $\int\left(12+x-x^{2}\right) \mathrm{d} x$

(iii) Showing all your working, find the area of the shaded region. $[4]$

2. (CIE 0606/2018/w/21/q8)

The diagram shows part of the curve $y=x+\mathrm{e}^{5-2 x}$, the normal to the curve at the point $A$ and the line $x=5$. The normal to the curve at $A$ meets the $y$-axis at the point $B$. The $x$-coordinate of $A$ is $2.5$.

(i) Find the equation of the normal $A B$.

(ii) Showing all your working, find the area of the shaded region.

3. (CIE 0606/2018/w/22/q9)

The diagram shows part of the curve $y=2 \sqrt{x}$. The normal to the curve at the point $A(4,4)$ meets the $x$-axis at the point $B$.

(i) Find the equation of the line $A B$.

(ii) Find the coordinates of $B$. [1]

(iii) Showing all your working, find the area of the shaded region. [4]

4. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 11 / \mathrm{q} 10)$

The diagram shows part of the graph of $y=2+\cos 3 x$ and the straight line $y=1.5$. Find the exact area of the shaded region bounded by the curve and the straight line. You must show all your working.

5. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 11 / \mathrm{q} 12)$

A curve is such that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 \sin \left(x+\frac{\pi}{3}\right)$. Given that the curve has a gradient of 5 at the point $\left(\frac{\pi}{3}, \frac{5 \pi}{3}\right)$, find the equation of the curve.

6. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 11 / \mathrm{q} 6)$

The diagram shows the curve $y=3 x^{2}-2 x+1$ and the straight line $y=2 x+5$ intersecting at the points $P$ and $Q$. Showing all your working, find the area of the shaded region.

7. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 12 / \mathrm{q} 9)$

The diagram shows the curve $y=4+2 \cos 3 x$ intersecting the line $y=5$ at the points $P$ and $Q$

(i) Find, in terms of $\pi$, the $x$-coordinate of $P$ and of $Q$. $[3]$

(ii) Find the exact area of the shaded region. You must show all your working.

8. $(\mathrm{CIE} 0606 / 2019 / \mathrm{m} / 22 / \mathrm{q} 10)$

The diagram shows the curve $y=1+x+5 \sqrt{x}$ and the straight line $y-3 x=3$. The curve and line intersect at the points $A$ and $B$. The lines $B C$ and $A D$ are perpendicular to the $x$-axis.

(i) Using the substitution $u^{2}=x$, or otherwise, find the coordinates of $A$ and of $B$. You must show all your working. $[6]$

(ii) Find the area of the shaded region, showing all your working.

9. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 23 / \mathrm{q} 7)$

The diagram shows part of the curve $y=x+\frac{6}{(3 x+2)^{2}}$ and the line $x=2$,

(i) Find, correct to 2 decimal places, the coordinates of the stationary point.

(ii) Find the area of the shaded region, showing all your working. $[4]$

10. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 23 / \mathrm{q} 9)$

The diagram shows the curve $y=16-x^{2}$ and the straight line $y=7$. Find the area of the shaded region. You must show all your working.

11.(CIE 0606/2020/s/12/q6)

The diagram shows the straight line $2 x+y=-5$ and part of the curve $x y+3=0 .$ The straight line intersects the $x$-axis at the point $A$ and intersects the curve at the point $B$. The point $C$ lies on the curve. The point $D$ has coordinates $(1,0)$. The line $C D$ is parallel to the $y$-axis.

(a) Find the coordinates of each of the points $A$ and $B$.

(b) Find the area of the shaded region, giving your answer in the form $p+\ln q$, where $p$ and $q$ are positive integers.

12. (CIE 0606/2020/w/13/q10)

(a) Show that $\frac{1}{x+1}+\frac{2}{3 x+10}$ can be written as $\frac{5 x+12}{3 x^{2}+13 x+10}$.

(b)

The diagram shows part of the curve $y=\frac{5 x+12}{3 x^{2}+13 x+10}$, the line $x=2$ and a straight line of gradient $1 .$ The curve intersects the $y$-axis at the point $P$. The line of gradient 1 passes through $P$ and intersects the $x$-axis at the point $Q$. Find the area of the shaded region, giving your answer in the form $a+\frac{2}{3} \ln (b \sqrt{3})$, where $a$ and $b$ are constants. $[9]$

13. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 13 / \mathrm{q} 9)$

The diagram shows part of the curve $x y=2$ intersecting the straight line $y=5 x-3$ at the point $A$. The straight line meets the $x$-axis at the point $B$. The point $C$ lies on the $x$-axis and the point $D$ lies on the curve such that the line $C D$ has equation $x=3$. Find the exact area of the shaded region, giving your answer in the form $p+\ln q$, where $p$ and $q$ are constants.

[8]

14. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 21 / \mathrm{q} 10)$

The diagram shows part of the graphs of $y=4 x^{\frac{2}{3}}$ and $y=(x-3)^{2}$. The graph of $y=(x-3)^{2}$ meets the $x$-axis at the point $A(a, 0)$ and the two graphs intersect at the point $B(b, 4)$.

(a) Find the value of $a$ and of $b$. $[2]$

(b) Find the area of the shaded region. $[5]$

Answer

1. (i) $A=(-2,6), B=(2,10)$

(ii) $12 x+\frac{x^{2}}{2}-\frac{x^{3}}{3}(+c)$

(iii) $A=\frac{32}{3}$

2. (i) $y=x+1$

(ii) $A=15.5$

3. (i) $y=-2 x+12$

(ii) $B=(6,0)$

(iii) $A=14 \frac{2}{3}$

4. Area $=\frac{\sqrt{3}}{3}-\frac{\pi}{9}$

5. $y=-2 \sin \left(x+\frac{\pi}{3}\right)+4 x+\frac{\pi}{3}+\sqrt{3}$

6. $A=\frac{256}{27}$

7. (i) $\frac{\pi}{9},-\frac{\pi}{9}$

(ii) $A=\frac{2 \sqrt{3}}{3}-\frac{2 \pi}{9}$

8. (i) $A(.25,3.75) B(4,15)$

(ii) $2.81$

9. (i) $x=0.43, y=0.98$

(ii) $A=2.75$

10. Area $=\frac{148}{3} 12345678910$

11. (a) $A=\left(-\frac{5}{2}, 0\right), B=\left(\frac{1}{2},-6\right)$

(b) Area $=9+\ln 8$

12. (a) Proof

(b) $\frac{18}{25}+\frac{2}{3} \ln \left(\frac{24}{5} \sqrt{3}\right)$

13. Total area $=\frac{2}{5}+\ln q$

14. (a) $a=3, b=1$ (b) $5.07$