Integration as the limit of a sumAQA A-Level Maths: Subtopic test

1. 1
   The area under the curve $y=x^2$ for $0\le x\le3$ is approximated by three rectangles, each of width 1, whose heights are the values of $y$ at the left-hand end of each strip.

   (a)

   Find the total area of the three rectangles.

   [1 mark]

   • A$5$
   • B$14$
   • C$9$
   • D$4$

   (b)

   Which statement about the total area of the rectangles is correct?

   [1 mark]

   • AIt is an overestimate, because $x^2$ is increasing so each rectangle reaches above the curve.
   • BIt is an underestimate, because $x^2$ is increasing so each rectangle lies below the curve.
   • CIt equals the exact area, because the heights are values of the function.
   • DNo comparison can be made without knowing the exact area.

   (c)

   Find, as a percentage of the exact area, the error in the approximation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For $1\le x\le4$ let $f(x)=2x+1$, and consider the sum $\sum_{x=1}^{4}(2x+1)\,\delta x$, where $\delta x$ is the small positive width of each strip.

   (a)

   Which definite integral is equal to the limit of the sum as $\delta x\to0$?

   [1 mark]

   • A$\int_0^4(2x+1)\,dx$
   • B$\int_1^4(x^2+x)\,dx$
   • C$\int_1^4(2x+1)\,dx$
   • D$\int_1^5(2x+1)\,dx$

   (b)

   Find the value of $\lim_{\delta x\to0}\sum_{x=1}^{4}(2x+1)\,\delta x$.

   [1 mark]

   • A$20$
   • B$22$
   • C$15$
   • D$18$

   (c)

   Explain why this limit gives the area under the graph of $y=f(x)$ between $x=1$ and $x=4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The region under the line $y=3x$ for $0\le x\le2$ is divided into $n$ strips of equal width. A rectangle is drawn on each strip with height equal to the value of $y$ at the right-hand end of the strip.

   (a)

   Show that the total area of the $n$ rectangles is $6+\frac{6}{n}$.

   [3 marks]

   (b)

   As $n\to\infty$ the total area of the rectangles tends to the exact area. Write down this limit and hence find the least $n$ for which the rectangles overestimate the exact area by less than $1\%$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A straight rod lies along the $x$-axis from $x=0$ to $x=4$, where $x$ is the distance in metres from one end. The density of the rod at distance $x$ is $\rho(x)=2+\frac{x}{2}$ kg per metre. The rod is thought of as many short pieces, each of length $\delta x$.

   (a)

   Explain how the total mass of the rod can be written as a limit of a sum, write it as a definite integral and find its value.

   [6 marks]

   (b)

   The rod is cut at $x=k$, where $0<k<4$, so that the piece from $x=0$ to $x=k$ has mass $5$ kg. Find $k$, justifying your choice of solution.

   [6 marks]

   Total for question 4: 12 marks

End of questions