Evaluate
4 √(ln(9-x)) dx
∫ ------------------------
2 √(ln(9-x)) + √(ln(x+3))
In pre-calculus we study different functions. As strange as this one is, the various part are all recognizable (except the integral). Call the function f(x).
It's kind of neat looking. We can see the domain is -2<=x<=8. We can easily calculate f(-2)=1, f(8)=0, and right in the middle f(3)=1/2
Ignoring the integral calculus, I'd just give them the function and say we want to find the area bounded by f(x), the x-axis, x=2 and x=4.
Looking at this region it looks quite a but like a trapezoid with height 2 and average base 1/2. We know it's curved but maybe we can use symmetry. Can we show it's half of the larger rectangle bounded up higher by y=1.
Lets translate it so (3,1/2) is at the origin
g(x)=f(x+3)-1/2
https://www.desmos.com/calculator/jdr59d8pg8
If this has origin symmetry we are done. Rather than show the work here. The Desmos graph has the algebra and indeed g(-x)=-g(x). The area between g, y=-1/2, x=-1 and x=1 is half of the blue bounding rectangle and thus equals 1.
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Posted by Jer
on 2025-02-07 11:49:39 |
Pre-calculus solution
