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If = 02 3 2 (x2+4 )dx+ 24 2x-4 dx, then 2 is equal to .
192
Let f(x)= 2(x2+4), x [0,2 3]: increasing, f(0)=2, f(2 3)= 216=4.
y= 2(x2+4) x= 2y-4, so f-1(y)= 2y-4 on [2,4].
For an increasing f: ab f(x) dx+ f(a)f(b)f-1(y) dy=b f(b)-a f(a)
(the two areas together fill the rectangle up to (b,f(b)) minus the one up to (a,f(a))).
=2 3 4-0 2=8 3
∴ 2=192
Answer: 192
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