Despite “appropriate use of technology” being a key part of the A-level syllabus, it is apparently Still A Thing to have to perform the trapezium rule by hand (rather than, say, asking Wolfram|Alpha like a normal mathematician would).
But still. If it’s in the syllabus, I suppose you can’t simply state a moral objection to the question and expect any marks, so you’re stuck with learning it. And you’re probably stuck with making the same errors as everyone else. Here are some questions to ask yourself if your answer is off:
1. Are you using radians?
I think this is probably the biggest source of errors in trapezium rule questions involving trig functions. Calculus, even numerical calculus, is always in radians. If your calculator isn’t, you will get the wrong answer.
2. Have you got the right number of strips and ordinates?
The question might ask for a particular number of strips or a particular number of ordinates (the values either side of each strip). If you have $n$ strips, you have $n+1$ ordinates. You always divide the width of the domain you’re interested by the number of strips, not the number of ordinates – imagine you just had one strip; then you’d have to divide the domain by one to get the height of the trapezium.
3. Have you got the multipliers right?
The $y$-value of the first and last ordinates are multiplied by 1; the others are multiplied by 2, and the whole lot added up then multiplied by half the strip-width. It’s easy to mess this up; I recommend using a table of values to keep track.
4. Have you multiplied by half the strip width?
Again, this step is very easy to miss. I suggest coming up with a ballpark answer before you do the calculation – just eyeball it to see what range of answers you’d be prepared to believe.
5. Have you accounted for positive and negative $y$-values?
It’s unusual for them to pull a sneaky trick like this, but not beyond the scope: make sure you’re looking out for it. If your curve drops below the $x$-axis, some or all of the result of the integration will be negative. If you’re looking for the area, you will need to account for this.
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