Calculus: Early Transcendentals, Chapter 5, 5.2, Section 5.2, Problem 57

You need to use the mean value theorem such that:
int_a^b f(x)dx = (b-a)f(c), c in (a,b)
int_(-1)^1 sqrt(1+x^2)dx = (1+1)f(c) = 2f(c)
You need to verify the monotony of the function f(x) = sqrt(1+x^2), such that:
f'(x) = x/(sqrt(1+x^2))
Since the function is even, then int_(-1)^1 sqrt(1+x^2)dx = 2int_0^1 sqrt(1+x^2)dx . Notice that f(x) increases on (0,1).
Hence, if 0Evaluate f(0) = sqrt 1 and f(1) = sqrt2 , such that:
sqrt1 Multiply by 2:
2<2f(c)<2sqrt2
Replace 2f(c) by int_(-1)^1 sqrt(1+x^2)dx such that:
2Hence, the inequality is verified using mean value theorem, without evaluating the integral.