Maths question help as level edexcel

9

For question 9b could someone explain to me why they please subtracted the linear equation from the equation of the curve and integrated that within the limits. I understand how to do the integration but I don’t get why they subtracted it.

https://imgur.com/a/f7g2JcS

Reply 1

mqb2766

21

Original post

by saminasb3

For question 9b could someone explain to me why they please subtracted the linear equation from the equation of the curve and integrated that within the limits. I understand how to do the integration but I don’t get why they subtracted it.
https://imgur.com/a/f7g2JcS

you want the yellow area which is the integral of the "vertical height" between the line and the curve. So the vertical height is curve - line

Reply 2

username79352

OP

9

So could I just do that method for any curve where the area we want is between the curve and line regardless of whether the curve is positive or negative?

Reply 3

mqb2766

21

Original post

by saminasb3

So could I just do that method for any curve where the area we want is between the curve and line regardless of whether the curve is positive or negative?

Sort of, though integration can give negative values when a curve is below the x-axis, so you just have to be aware of that. Here its easy as the curve > line for all x, so
Integral of curve - line
will give the yellow area (positive value). Its also worth noting that you could write it as (sketch what youre doing)
integral of curve - integral of line
as both integrals give the relevant area between the curves and the x-axis, so the difference is what you want and equivalent to the first way.

If the curves crossed and you wanted the total area, youd need to split the integral as appropriate (crossing points) and integrate the appropriate difference over the relevant domains, always ensuring that the vertical height difference was positive.

Another way to think about it is to write it as
integral of |curve1 - curve2|
then think about how to remove the absolute value function (split the integral / flip the difference) to ensure that the integrand is always positive.

(edited 1 year ago)

Reply 4

username79352

OP

9

Thank you for the help! I knew that I could split the areas into bits and subtract areas of shapes to get the yellow area but I never knew I could subtract the equations. The book never mentioned anything about that.

Reply 5

mqb2766

21

Original post

by saminasb3

Thank you for the help! I knew that I could split the areas into bits and subtract areas of shapes to get the yellow area but I never knew I could subtract the equations. The book never mentioned anything about that.

Its a fairly common way to view it so either as the difference in areas (integrate each curve, then subtract) or integrate the vertical difference between them (subtract the equations then integrate) as subracting the equations gives the vertical difference for a given value of x which is the usual view of integration.

Maths question help as level edexcel

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