Area between curves and integration as the limit of a sumEdexcel A-Level Maths: Subtopic test

1. 1
   The curves $C_1$ and $C_2$ have equations $y=x^2-5x+6$ and $y=4-x^2$ respectively. They intersect at two points.

   (a)

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

   [1 mark]

   • A$x=\frac12$ and $x=-2$
   • B$x=2$ and $x=3$
   • C$x=-\frac12$ and $x=-2$
   • D$x=\frac12$ and $x=2$

   (b)

   Between the points of intersection $C_2$ is above $C_1$. Find the vertical distance between the curves at a general value of $x$ in this interval.

   [1 mark]

   • A$2x^2-5x+2$
   • B$-2x^2+5x+10$
   • C$-2x^2+5x-2$
   • D$5x-2$

   (c)

   Find the area of the region enclosed between the two curves.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has parametric equations $x=t^2$, $y=3t$, for $t\ge0$.

   (a)

   Find $\frac{\mathrm{d}x}{\mathrm{d}t}$.

   [1 mark]

   • A$2t$
   • B$3$
   • C$\frac{1}{2t}$
   • D$t^2$

   (b)

   Which integral gives the area between $C$, the $x$-axis and the lines $x=0$ and $x=4$?

   [1 mark]

   • A$\int_0^4 6t^2\,\mathrm{d}t$
   • B$\int_0^2 6t^2\,\mathrm{d}t$
   • C$\int_0^2 3t\,\mathrm{d}t$
   • D$\int_0^4 3t\,\mathrm{d}t$

   (c)

   Find the area described in part (b).

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The area under the curve $y=x^2$ for $0\le x\le3$ is being investigated using rectangles of equal width.

   (a)

   Three rectangles of width $1$ are drawn, each with height equal to the value of $y$ at the left-hand end of its strip. Calculate their total area, and state with a reason whether it is an underestimate or an overestimate of the area under the curve.

   [3 marks]

   (b)

   The width of each rectangle is made smaller and smaller. Write the exact area under the curve as the limit of a sum, express it as a definite integral and evaluate it. Hence find by how much your answer to (a) is too small.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has parametric equations $x=4\sin t$, $y=3\cos t$, for $0\le t\le\frac{\pi}{2}$. The region $R$ is bounded by $C$ and the coordinate axes.

   (a)

   Show that the area of $R$ is $3\pi$.

   [6 marks]

   (b)

   The straight line $L$ passes through $(0,3)$ and $(4,0)$.
   (i) Find the equation of $L$.
   (ii) Show that the point on $C$ where $t=\frac{\pi}{4}$ lies above $L$.
   (iii) Hence find the exact area of the region between $C$ and $L$, given that $C$ lies above $L$ throughout.

   [6 marks]

   Total for question 4: 12 marks

End of questions