Total geometric areas using integration

Question

Given the function f(x)=x−7, find the total area between f(x) and the x-axis over the interval [−7,9].

Answer 1

While the actual integration process is basic some consideration must be given to the concept of "total area".

Sketching the graph a portion from x = -7 to x = 7 is below the x axis and is considered negative by integration.

Therefore the total area by integration is not the same as area by geometry.

By geometry the area of one triangle from -7 to 7 is 98 and the area of the other from 7 to 9 is 2.

This is a total of 100./2

The integration is done is two parts

(x²/2) - 7x evaluated from -7 to 7 (this is the "negative" area)

(49/2) - 49 - [(49/2) + 49}

-98

(x²/2) - 7x evaluated from 7 to 9 (this is the "positrive" area)

(81/2) - 63 - [(49/2) - 49]

2

Q.E.D.

Answer 2

it's the area of a triangle with base = (9--7)= 16

and height = f(9)-f(7) = 2--14=16

Area = bh/2 = 16(16)/2 = 8(16= 128

it may help to sketch a rough graph of the line and construct the triangle

with its 3 vertices (-7, f(-7)) =(-7,-14), (9,f(9))= (9,2) and (9,-14)

distance from (-7,-14) to (9,-14) = base=16

distance from (9,2) to (9,-14) = height= 16

or integrate x-7

= x^2/2 -7x

evaluate from -7 to 9

but not this way

= 81/2 -56 - (49/2+49/2)

= 40.5 -56 +49

= 33.5 which is the wrong answer

instead find where the line crosses the x axis

which is where x=7

evaluate from -7 to 7 and then from 7 to 9

49/2 -49 -(49/2 +49) + 49/2+49 -(81/2 -63

= 49 +130/2 -49+63

=65 +63

=128

if you work the problem two different ways and get the same answer, that's

virtually certain to be correct

working the problem by geometry or by calculus gives the identically same answer,

as it has to be, if you work the problem correctly.

Areas are never negative, except in the twilight zone or an alternative universe

when you integrate, put the higher line 1st then subtract the lower line, to avoid negative "areas"

IF for some reason this problem was intended to be x^2-7 instead of x-7

then the Total Area = 223 1/3 + 28sqr7/3

= about 248.0270122

solved only by calculus, not geometry

integrating x^2-7

= x^3/3 -7x

evaluated 3 times, from

-7 to -sqr7, from -sqr7 to sqr7 and from sqr7 to 9