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jennychrin95

18.04.2021

Mathematics

Let's divide the shaded region into two areas:

area 1: x = 0 ---> x = 2

ares 2: x = 2 ---> x = 4

In area 1, we need to find the area under g(x) = x and in area 2, we need to find the area between g(x) = x and f(x) = (x - 2)^2. Now let's set up the integrals needed to find the areas.

Area 1:

$A\frac{}{1} = ∫g(x)dx = ∫xdx = \frac{1}{2} {x}^{2} | \frac{2}{0} = 2$

Area 2:

$A\frac{}{2} = ∫(g(x) - f(x))dx$

$= ∫(x - {(x - 2)}^{2} )dx$

$= ∫( - {x}^{2} + 5x - 4)dx$

$= ( - \frac{1}{3}{x}^{3} + \frac{5}{2} {x}^{2} - 4x) | \frac{4}{2}$

= 2.67 - ( - 0.67) = 3.34

Therefore, the area of the shaded portion of the graph is

A = A1 + A2 = 5.34

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