Integration as the limit of a sumAQA A-Level Maths: Mind map

Exam questions on Integration as the limit of a sum

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.
   Find, as a percentage of the exact area, the error in the approximation.2 marks
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.
   Explain why this limit gives the area under the graph of $y=f(x)$ between $x=1$ and $x=4$.2 marks
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.
   Show that the total area of the $n$ rectangles is $6+\frac{6}{n}$.3 marks