What is “area under the curve?”

• The definite integral can be used to find the area between a graph curve and the ‘x’ axis, between two given ‘x’ values. This area is called the ‘area under the curve’ regardless of whether it is above or below the ‘x’ axis.

• When the area is above the x axis, the area is positive.

• When the area is below the x axis, the area is negative.

Example problem

• y = x² - x – 2• We need to create three

separate integrals

Finding the integrals

• The zeros of the function f(x) that lie between -2 and 3 form the boundaries of the separate area segments, and these will be our integrals

• -2 < x < -1 • -1 < x < 2• 2 < x < 3

In Integral Form:

• Integrate the equation using the general power rule:

• 1/3x^3 – 1/2 x^2 – 2x

• Plug in beginning points and endpoints for each of three intervals and subtract the beginning point from the end point.

Plug in numbers and solve

• 1/3(-1)^3 – ½ (-1)^2 – 2(-1) – (1/3(-2)^3 – ½ (-2)^2 – 2(-2) = 1.83333

• 1/3(2)^3 – ½ (2)^2 – 2(2) – (1/3(-1)^3 – ½ (-1)^2 – 2(-1)) = 4.5

• 1/3(3)^3 – ½ (3)^2 – 2(3) – (1/3(2)^3 – ½ (2)^2 – 2(2)) = 1.83333

• The total shaded area will be A = A1 + A2 + A3

• Total area = 1.83333 + 4.5 + 1.83333 = 8.16666

• The net area takes into account the negative area:

• 1.83333 - 4.5 + 1.83333 • = - 0.83333