Chapter 4: 46 (page 386)

$f (x) = x^{2} - x - 1, g (x) = 5 - x^{2}, [- 2, 3]$

Short Answer

Expert verified

The exact area between the graphs is $\frac{9}{2}$ or $4.5$ square units.

Step by step solution

Given information

$f (x) = x^{2} - x - 1, g (x) = 5 - x^{2}, [- 2, 3]$

Calculation

Consider the following functions:

$f (x) = x^{2}; g (x) = x + 2 .$

The objective is to find the exact area of the region between the two graphs on the interval $[- 3, 3]$ .

The point of intersection of the two function is,

x^{2} = x + 2 x^{2} - x - 2 = 0 x^{2} - 2 x + x - 2 = 0 x (x - 2) + 1 (x - 2) = 0 (x - 2) (x + 1) = 0 x = - 1 o r x = 2

The graph is show below,

The exact area of the region is calculated as follows:

role="math" localid="1651079495478" $\int_{- 1}^{2} | f (x) - g (x) | d x = \int_{- 1}^{2} ((x + 2) - x^{2}) d x = \int_{- 1}^{2} (x + 2 - x^{2}) d x = {(\frac{x^{2}}{2} + 2 x - \frac{x^{3}}{3})}_{- 1}^{2} = (\frac{2^{2}}{2} + 2 (2) - \frac{2^{3}}{3}) - (\frac{(- 1)^{2}}{2} + 2 (- 1) - \frac{(- 1)^{3}}{3})$

$= \frac{9}{2}$ or $4.5$ square units.

Therefore, the exact area between the graphs is $\frac{9}{2}$ or $4.5$ square units

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$f (x) = x^{2} - x - 1, g (x) = 5 - x^{2}, [- 2, 3]$

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f(x)=x2-x-1,g(x)=5-x2,[-2,3]

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$f (x) = x^{2} - x - 1, g (x) = 5 - x^{2}, [- 2, 3]$