Area between curves

Question

y = x³ + 3x and y = 4x

Determine its entire area in units²

^{Area =}

Answer 1

1/2 or 0.5 square units

find where the two lines intersect

set their 7 values equal

x^3 +3x = 4x

x^3 -x = 0

x(x^2 -1) = 0

set the factors =0

x^2 = 1, x= + or -1, x=0

x=0, -1, +1

(0,0), (0,4) and (0,-4) are the points where the two lines intersect

the areas between the curves are two equal areas, one in quadrant I, another in quadrant III

calculate the area in quadrant I, then multiply it by 2

A = integral of 4x - x^3 -3x = integral of x -x^3 = x^2/2 - x^4/4

evaluate at x= 1, to get 1/2 -1/4 = 1/4

multiply it by 2

2A = 1/2 = area between the two lines

it helps to sketch a graph of the line and the cubic function

then you'll realize you can't use limits of integration -1 to 1, as then area =0, as the negative area in quadrant III cancels the positive area in quadrant I

How do you feel when you hear people say "Let's Go Brandon"?

Answer 2

y = x³ + 3x and y = 4x

Area = ∫_a^b [f(x)−g(x)]dx (Integration)

Area = 1/2 units²