Area of one strip of width \delta x at position x?

About f(x)\,\delta x (a rectangle of height f(x)).

What does \delta x\to0 achieve?

The strips become thinner, so the rectangle approximation becomes exact.

Left-hand rectangles on an increasing curve: over or under?

An underestimate, because each rectangle lies below the curve.

Right-hand rectangles on y=3x, 0\le x\le2: total area?

6+\frac6n, which tends to 6 as n\to\infty.

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

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Write integration as a limit of a sum.: $\int_a^bf(x)dx=\lim_{\delta x\to0}\sum_{x=a}^bf(x)\,\delta x$
Area of one strip of width $\delta x$ at position $x$ ?: About $f(x)\,\delta x$ (a rectangle of height $f(x)$ ).
What does $\delta x\to0$ achieve?: The strips become thinner, so the rectangle approximation becomes exact.
Left-hand rectangles on an increasing curve: over or under?: An underestimate, because each rectangle lies below the curve.
Right-hand rectangles on an increasing curve: over or under?: An overestimate.
Left-hand rectangles on a decreasing curve?: An overestimate.
Convert $\lim\sum_{x=1}^{4}(2x+1)\delta x$ to an integral.: $\int_1^4(2x+1)dx=18$
What does the symbol $\int$ stand for?: A stretched S for 'sum'.
Width of each strip for $n$ equal strips on $[a,b]$ ?: $\frac{b-a}{n}$
Value of $\sum_{r=1}^nr$ ?: $\frac{n(n+1)}{2}$
Right-hand rectangles on $y=3x$ , $0\le x\le2$ : total area?: $6+\frac6n$ , which tends to $6$ as $n\to\infty$ .
Mass of a short piece of rod of density $\rho(x)$ and length $\delta x$ ?: About $\rho(x)\,\delta x$ .

Exam questions on Integration as the limit of a sum

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
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
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

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).