Hey was looking for help with a couple questions ive been stuck on
Part ii for the area under the curve questions
Thanks!
Hey Deng!!
Interestingly, I'd actually approach that first question not as an integration area question, but with some geometry! Let's have a think. The area we want is actually just the area of the triangle enclosed by the axes and the tangent, then subtract the area of the quarter circle! Let me know if that's unclear what I mean there!
So, we need the intercepts of the tangent so we can find the area of that triangle. Use whatever method you like, I'll use substitution:
So the triangle has a width of 5 and a height of 15/4, let's use that to find the area we need:
I'll let you take it from there
For Question 2, we can massively simplify that expression by recognising that sine squared plus cosine squared of the same value is just 1. So, using that Pythagorean Identity, the expression becomes:
Now we just need to think about this intuitively (easier than rigorous mathematics). We want the smallest value of this expression over the given domain, which means we want the denominator as large as possible. Well, the maximum value of sin^2 is 1, so that means the
minimum value of the expression is:
Part (ii) of that last question is a little tricky! Basically, we can get it by understanding that
the integral of the first derivative will get us back to the function itself!
I'll leave you to have the full attempt. But let me get you started. So, from your sketch in Part (i), you know that the graph of f'(x) has intercepts at x=-2, x=-1, and x=3. First, let's find the area enclosed between x=3 and x=-1. Use our regular method, and remember, we are integrating a derivative! The result will be our original function:
We now just need to evaluate this at 3 and -1, which we can do, because we are given these values in the graph:
The same principle would apply to the other part of the area between x=-1 and x=-2, but with an absolute value sign as is normal for these sorts of questions
then just add the two areas together to get your answer!
That last one was a bit of a doozy, does that make sense Deng?