Expert: Socrates - 9/26/2006

Question
I am working through a study guide and this one is stumping me.

Q: Suppose f is continuous and x^2 <= f(x) <= 6 for all x (is an element of) [-1,2].  Find values A and B such that A <= (integration symbol with 2 as upper and -1 as lower values) f(x)dx <= B.

Thanks for your help in advance.

Answer
Since x^2 <= f(x) <= 6  when x is between -1 and 2 ,

S x^2 dx <= S f(x)dx <= S 6 dx ,

where the limits for each of the integrals are from -1 to 2

But it is easy to evaluate the two integrals on the outside

S x^2 dx = (1/3)2^3  - (1/3)(-1)^3 = 9/3 = 3

S 6 dx = (6)(2) - (6)(-1) = 18


So now we know that

3 <= S f(x)dx <= 18

and A = 3 , B = 18