1. ## Definite integral problem

The diagram above shows the curve $y=2x^2-5x+5$ and straight line $y=5-x$.Find

(a)the points if intersection of the two graphs

i get the answer $(0,5)(2,3)$

(b)The area of the shaded region.

2. I've never really solved a problem like this before (or any integration problem actually) but would this work:

$\int_{0}^{3} 5-x\ dx-\left(\int_{0}^{3}5-x\ dx\ -\ \int_{0}^{3}2x^2-5x+5\ dx \right )$

Then do a similar thing for $x\in \left [ 3,5\right ]$ and add the results together.

3. this is my step to get the answer but the answer provided not same with my answer

$\frac{1}{2}(5)(5) - \left(\int_{0}^{5}(2x^2-5x=5) dx$
$=\frac{25}{2}-(\frac 2{x^3}{3}-\frac{5x^2}{2})_0^5$
$=\frac{25}{2}-({2(5)^3}{3}-\frac{5(2)^2}{2}=5(5))-0$
$=\frac{25}{2}-(\frac{275}{6}$
$=\frac{-33}{1}{3}$

is that correct??lol,my latex is wrong

4. You can do this way.
[Intg(5 - x)*dx between 0 to 5 -[ Inte(2x^2 -5x + 5)*dx] between 0 to 3.