Total geometric areas

Question

Find the total area between the curve of the function f(x)=2x+4 and the x-axis over the interval [−9,−1].

Answer 1

F(x) = x² + 4x

(-1)² + 4(-1) - [(-9)² + 4(-9)]

Do the arithmetic and answer.

Answer 2

it helps to sketch the graph of the two lines

on is y=0, the x axis, a horizontal line

the other is an upward sloping straight line, with slope = 2

also draw 2 vertical lines

x=-9 and x=-1

calculate the area of the two triangles formed

one small triangle above the x axis has area = bh/2 = (1)(2)/2 = 1

the larger triangle below the x axis has area = bh/2 = (7)(14)/2 = 7(7) = 49

the two given lines, y=2x+4 and y=0 (the x axis) intersect

at 2x+4 = 0

2x=-4

x =-2

y =0

the point (-2,0)

the total area of the two triangles = 1 + 49= 50

that's the geometry method of solving the problem

you could also use calculus

and integrate 2x+4 and integrate -2x-4

= x^2 +4x

evaluate respectively from -9 to -2 and from -2 to -1

= (-9)^2 +4(-9) - [(-2)^2 +4(-2)] + [(-2)^2 +4(-2)] - [(-1)^2 +4(-1)]

= 81-36+ 4+4 -3

= 45+8-3

= 50

if you get the identical answer working the problem two different ways

you're virtually certain it's correct