# Thread: need help integral theorems

1. ## need help integral theorems

State true and false
1>If f and g are contineous and f(x)>= g(x) for a<x<b, then integral(a to b)f(x)dx>=integral(a tob)g(x)dx
2>If f and g are contineous and f(x)>= g(x) for a<x<b, then f '(x)>=g'(x) for a<x<b

2. #1 is a theorem. You should see if you could prove it. It is actually very important.

#2: On [0,1] consider $f(x) = 2 - x^2 \quad \& \quad g(x) = x^2 .$

3. Hello, Gracy!

State true and false.

(1) If $f$ and $g$ are continuous and $f(x) \geq g(x)$ for $a,
then: . $\int^b_a f(x)\,dx\:\geq \:\int^b_ag(x)\,dx$

I would say True . . .
Code:
|   *
|   :*        :
|   :  *      :
|   :     *   :
|   :         * f(x)
|   :         :
|   :         o g(x)
|   :      o  :
|   :  o      :
|   o         :
--+---+---------+--
|   a         b

If $f(x)$ is always "above" $g(x)$ on $(a,\,b)$
. . then the area under $f(x)$ will be $\geq$ the area under $g(x).$

(2) If $f$ and $g$ are continuous and $f(x) \geq g(x)$ for $a,
then: . $f '(x) \geq g'(x)$ for $a

This is False . . .
Code:
|   *
|   :*        :
|   :  *      :
|   :     *   :
|   :         * f(x)
|   :         :
|   :         o g(x)
|   :      o  :
|   :  o      :
|   o         :
--+---+---------+--
|   a         b

While $f(x)$ is above $g(x)$, we see that
. . $f'(x)$ is negative while $g'(x)$ is positive.