Please help (Integral Calculus question)

Find the area bounded by the curve $y = 5x - x^2$ and the line $y = x$

How do I do this??

Question

Please help (Integral Calculus question)

Find the area bounded by the curve $y = 5x - x^2$ and the line $y = x$

How do I do this??

anonymous · Answer

k I'm taking this as I go so give me a second first you need to find the points of intersection between the two lines do you know how to do this?

lgbasallote · Answer

no...how?

lgbasallote · Answer

well it is given in my book...but i'd like to know as well how to find it without its being given

anonymous · Answer

don't you just set the equations equal to one another and then solve? because basically you want to know what value of x gives the same value of y for both equations right?

anonymous · Answer

well what values I should say

lgbasallote · Answer

ahh so substitution method...why didn't i think of that

$$x = 5x - x^2$$
$$0 = 4x - x^2$$
$$0 = x(4 - x)$$
x =0
x = 4

these are the intersections right?

so point (0,0) and (4,4)....exactly what it says in my book thanks

lgbasallote · Answer

so what's next after the points of intersection?

anonymous · Answer

okay you need to use a formula there aren't different integration formulas for different situations but this one was straight forward luckily. First for the interval x= 0 to 4 what function is bigger?

anonymous · Answer

there are different** sorry typo

lgbasallote · Answer

well seems y = 5x - x^2

anonymous · Answer

did you check? I haven't done the math but you should probably check do you know how to?

anonymous · Answer

It is really easy you just have to put any value of x in btw 0 and 4 into both equations and see which is bigger I used 1 and it turns out you were right haha it was pretty obvious but when both the equations are exponential it won't be so obvious.

lgbasallote · Answer

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that's the  graph