Integration (AS)AQA A-Level Maths: Topic test

1. 1
   A curve $C$ has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=4x^{3}-\dfrac{3}{\sqrt{x}}$ for $x>0$, and $C$ passes through the point $(1,5)$.

   (a)

   Which expression is $\displaystyle\int\left(4x^{3}-\dfrac{3}{\sqrt{x}}\right)\mathrm{d}x$?

   [1 mark]

   • A$x^{4}-3x^{1/2}+c$
   • B$x^{4}-6x^{1/2}+c$
   • C$12x^{2}+\frac32x^{-3/2}+c$
   • D$x^{4}+6x^{1/2}+c$

   (b)

   What is the value of the constant of integration $c$?

   [1 mark]

   • A5
   • B0
   • C12
   • D10

   (c)

   Find the value of $y$ on $C$ when $x=4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The integral $I$ is defined by $I=\displaystyle\int_{1}^{4}\left(3\sqrt{x}-\dfrac{1}{x^{2}}\right)\mathrm{d}x$.

   (a)

   Which expression is an antiderivative of $3\sqrt{x}-\dfrac{1}{x^{2}}$?

   [1 mark]

   • A$2x^{3/2}+\frac1x$
   • B$2x^{3/2}-\frac1x$
   • C$\frac92x^{1/2}+2x^{-3}$
   • D$\frac23x^{3/2}+x^{-1}$

   (b)

   What is the value of $I$?

   [1 mark]

   • A$-\dfrac{53}{4}$
   • B$\dfrac{59}{4}$
   • C$\dfrac{53}{4}$
   • D$\dfrac{65}{4}$

   (c)

   Hence find the exact value of $\displaystyle\int_{1}^{4}\left(3\sqrt{x}-\dfrac{1}{x^{2}}+2\right)\mathrm{d}x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^{2}-4x+5$ and the line $L$ has equation $y=x+1$. They intersect at two points, and a finite region $R$ is enclosed between them.

   (a)

   Find the $x$-coordinates of the points where $C$ and $L$ intersect.

   [3 marks]

   (b)

   Find the area of $R$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=3x^{2}-16x+12$ and passes through the point $(1,5)$. $C$ crosses the $x$-axis at the origin and at two other points.

   (a)

   Find the equation of $C$ and show that it can be written as $y=x(x-2)(x-6)$.

   [6 marks]

   (b)

   Find the total area of the finite regions bounded by $C$ and the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The finite region $R$ is bounded by the curve $y=9-x^{2}$ and the $x$-axis.

   (a)

   Between which limits should $y=9-x^{2}$ be integrated to find the area of $R$?

   [1 mark]

   • A$0$ and $9$
   • B$-9$ and $9$
   • C$0$ and $3$
   • D$-3$ and $3$

   (b)

   What is the area of $R$?

   [1 mark]

   • A36
   • B18
   • C54
   • D72

   (c)

   The line $y=5$ cuts the curve at $x=\pm2$. Find the area of the part of $R$ that lies above the line $y=5$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{x^{5}+4}{x^{3}}$ for $x>0$.

   (a)

   Which expression is equal to $\dfrac{x^{5}+4}{x^{3}}$?

   [1 mark]

   • A$x^{2}+4$
   • B$x^{5/3}+4$
   • C$x^{2}+4x^{-3}$
   • D$x^{8}+4x^{3}$

   (b)

   Which expression is $y$, where $c$ is a constant?

   [1 mark]

   • A$\frac{x^{3}}{3}+2x^{-2}+c$
   • B$\frac{x^{3}}{3}-2x^{-2}+c$
   • C$\frac{x^{3}}{3}-2x^{-4}+c$
   • D$x^{3}-2x^{-2}+c$

   (c)

   The curve passes through the point $(1,1)$. Find the value of the constant of integration.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The real number $k$ is greater than $1$ and satisfies $\displaystyle\int_{1}^{k}\left(6x^{2}-2x\right)\mathrm{d}x=44$.

   (a)

   Show that $2k^{3}-k^{2}-45=0$.

   [3 marks]

   (b)

   Hence show that $k=3$ is the only possible value of $k$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=x^{3}-x^{2}$ and the line $L$ has equation $y=6x$. The curve and the line enclose two finite regions.

   (a)

   Find the coordinates of the three points where $C$ and $L$ intersect.

   [6 marks]

   (b)

   Find the total area of the two regions enclosed by $C$ and $L$.

   [6 marks]

   Total for question 8: 12 marks

End of questions