P2: IntegrationEdexcel International A Level Maths: Topic test

1. 1
   The curve $C$ has equation $y=\dfrac{8}{x^2}+x$ for $x>0$.

   (a)

   Find $\displaystyle\int\left(\dfrac{8}{x^2}+x\right)\mathrm{d}x$.

   [1 mark]

   • A$\dfrac{8}{x}+\dfrac{x^2}{2}+c$
   • B$-\dfrac{16}{x^3}+\dfrac{x^2}{2}+c$
   • C$-\dfrac{8}{x}+\dfrac{x^2}{2}+c$
   • D$-\dfrac{8}{3x^3}+\dfrac{x^2}{2}+c$

   (b)

   Find the value of $\displaystyle\int_1^4\left(\dfrac{8}{x^2}+x\right)\mathrm{d}x$.

   [1 mark]

   • A$\dfrac{27}{2}$
   • B$-\dfrac{27}{2}$
   • C$\dfrac{3}{2}$
   • D$6$

   (c)

   The finite region $R$ is bounded by $C$, the $x$-axis and the lines $x=2$ and $x=a$, where $a>2$. Find the area of $R$ in terms of $a$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The integral $I=\int_1^3\ln x\,\mathrm{d}x$ is to be estimated using the trapezium rule with four strips of equal width.

   (a)

   Which of the following lists the values of $x$ at which the $y$-ordinates are needed?

   [1 mark]

   • A$1,\ 2,\ 3$
   • B$1,\ 1.5,\ 2,\ 2.5,\ 3$
   • C$1,\ 1.25,\ 1.5,\ 1.75,\ 2$
   • D$1.5,\ 2,\ 2.5,\ 3$

   (b)

   Which of the following is the trapezium rule estimate of $I$, correct to $3$ decimal places?

   [1 mark]

   • A$2.564$
   • B$0.778$
   • C$1.242$
   • D$1.282$

   (c)

   State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of $I$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^2-5x+9$ and the line $l$ has equation $y=x+4$. The curve and the line intersect at two points.

   (a)

   Find the $x$-coordinates of the two points of intersection.

   [3 marks]

   (b)

   Find the area of the finite region bounded by $C$ and $l$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=\sqrt{x}$ for $x\geqslant0$ and the line $l$ has equation $y=\dfrac12x$. The curve and the line meet at the origin $O$ and at the point $A$.

   (a)

   Find the exact area of the finite region $R$ bounded by $C$ and $l$.

   [6 marks]

   (b)

   Use the trapezium rule with four strips of equal width to estimate $\int_0^4\sqrt{x}\,\mathrm{d}x$. Hence estimate the area of $R$, and state with a reason whether your estimate of the area of $R$ is less than or greater than the exact value found in part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $C$ has equation $y=3x^2-x^3$. It meets the $x$-axis at the origin $O$ and at the point $A(3,0)$, and $y\geqslant0$ for $0\leqslant x\leqslant3$.

   (a)

   Find $\displaystyle\int(3x^2-x^3)\,\mathrm{d}x$.

   [1 mark]

   • A$x^3-\dfrac{x^4}{4}+c$
   • B$6x-3x^2+c$
   • C$\dfrac{x^3}{3}-\dfrac{x^4}{4}+c$
   • D$x^3-x^4+c$

   (b)

   Find the area of the region bounded by $C$ and the $x$-axis.

   [1 mark]

   • A$27$
   • B$\dfrac{81}{4}$
   • C$\dfrac{27}{4}$
   • D$\dfrac{189}{4}$

   (c)

   Find the area of the region bounded by $C$, the $x$-axis and the lines $x=1$ and $x=3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=x^3$ and the line $l$ has equation $y=4x$. The curve and the line meet at three points.

   (a)

   Which of the following gives the $x$-coordinates of the three points of intersection?

   [1 mark]

   • A$0$ and $2$ only
   • B$-2,\ 0,\ 2$
   • C$-4,\ 0,\ 4$
   • D$0,\ \pm\sqrt2$

   (b)

   Find the total area of the two finite regions enclosed between $C$ and $l$.

   [1 mark]

   • A$0$
   • B$4$
   • C$16$
   • D$8$

   (c)

   Find the area of the region bounded by $C$, $l$, the $y$-axis and the line $x=1$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A surveyor measures the depth of a river at $2$ m intervals across its width of $12$ m. The depths, in metres, at distances $0,2,4,6,8,10,12$ m from one bank are $0,\ 1.2,\ 1.9,\ 2.2,\ 1.9,\ 1.2,\ 0$ respectively. The cross-sectional area of the river is the area under the depth curve.

   (a)

   Use the trapezium rule with all seven measured depths to estimate the cross-sectional area of the river.

   [3 marks]

   (b)

   The depth is also modelled by $y=0.06x(12-x)$, where $x$ metres is the distance from the bank. Use integration to find the cross-sectional area given by this model.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=9-x^2$ and the line $l$ has equation $y=x+3$. The curve and the line meet at the points $P$ and $Q$, and the finite region $R$ is bounded by $C$ and $l$.

   (a)

   Find the exact area of $R$.

   [6 marks]

   (b)

   Use the trapezium rule with five strips of equal width to estimate $\int_{-3}^{2}(9-x^2)\,\mathrm{d}x$. Hence estimate the area of $R$, and explain why your estimate is smaller than the exact area found in part (a).

   [6 marks]

   Total for question 8: 12 marks

End of questions