area x^2-2,0,-2,2 — step-by-step solution

1) The area between

x^{2} - 2

and

0

[-2, 2]

is the integral of the absolute value of their difference

Split the interval at the interior intersections:

(-2, - \sqrt{2}, \sqrt{2}, 2)

[-2, - \sqrt{2}]

x^{2} - 2

is above

0

: integrate

x^{2} - 2

⇥1.1) Let's integrate:

\int_{-2}^{- \sqrt{2}} (x^{2} - 2) d x

⇥ⅰ

\int (x^{2} - 2) d x

\int x^{2} d x - \int 2 d x

⇥

\int x^{2} d x - \int 2 d x

⇥ⓘ An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int x^{2} d x

\frac{x^{3}}{3}

■The derivative of

- 2 x

-2

, so

- 2 x

is an antiderivative of

-2

- \int 2 d x

- 2 x

▷

\frac{x^{3}}{3} - 2 x

⇥1.2) Let's evaluate over the interval

[-2, - \sqrt{2}]

(\frac{{(- \sqrt{2})}^{3}}{3} - 2 \cdot (- \sqrt{2})) - (\frac{{(-2)}^{3}}{3} - 2 \cdot (-2))

⇥ⓘ Powers are computed before multiplications, additions and subtractions. With an even integer exponent, the minus sign in the base disappears; with an odd integer exponent, it moves in front of the power:

{(- \sqrt{2})}^{3}

- {(\sqrt{2})}^{3}

■Extract the minus signs: an even number of minus signs cancels out, an odd number leaves one minus sign:

- 2 \cdot (- \sqrt{2})

2 \cdot \sqrt{2}

■A minus sign before a parenthesis changes the sign of each of its terms:

- (\frac{{(-2)}^{3}}{3} - 2 \cdot (-2))

- \frac{{(-2)}^{3}}{3} + 2 \cdot (-2)

⇥

- \frac{{(\sqrt{2})}^{3}}{3}

+ 2 \cdot \sqrt{2}

- \frac{{(-2)}^{3}}{3}

+ 2 \cdot (-2)

⇥ⓘ A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

{(\sqrt{2})}^{3}

2^{\frac{1}{2} \cdot 3}

■Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-2)}^{3}

(-2) \cdot (-2) \cdot (-2)

-8

■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-2)

-4

⇥

- \frac{2^{\frac{1}{2} \cdot 3}}{3}

+ 2 \cdot \sqrt{2}

- (\frac{-8}{3})

- 4

⇥ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{1}{2} \cdot 3

\frac{1 \cdot 3}{2}

\frac{3}{2}

■Two minus signs cancel out:

- (\frac{-8}{3})

\frac{8}{3}

⇥

- \frac{2^{\frac{3}{2}}}{3}

+ 2 \cdot \sqrt{2}

+ \frac{8}{3}

- 4

⇥ⓘ Powers are computed before multiplications, additions and subtractions:

2^{\frac{3}{2}}

2 \cdot \sqrt{2}

■To add fractions, put them over the same denominator, then add the numerators:

\frac{8}{3} - 4

\frac{8 - 4 \cdot 3}{3}

⇥

- \frac{2 \cdot \sqrt{2}}{3} + 2 \cdot \sqrt{2} + \frac{8 - 4 \cdot 3}{3}

⇥ⓘ Like terms are grouped by adding their coefficients:

- \frac{2 \cdot \sqrt{2}}{3} + 2 \cdot \sqrt{2}

\frac{4 \cdot \sqrt{2}}{3}

■Multiplications are done before additions and subtractions:

4 \cdot 3

12

⇥

\frac{4 \cdot \sqrt{2}}{3} + \frac{8 - 12}{3}

⇥ⓘ The numerator and the denominator are computed before the division:

8 - 12

-4

⇥

\frac{4 \cdot \sqrt{2}}{3} - \frac{4}{3}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{4 \cdot \sqrt{2}}{3} - \frac{4}{3}

\frac{4 \cdot \sqrt{2} - 4}{3}

⇥

\frac{4 \cdot \sqrt{2} - 4}{3}

The area on

[-2, - \sqrt{2}]

\frac{4 \cdot \sqrt{2} - 4}{3}

[- \sqrt{2}, \sqrt{2}]

0

is above

x^{2} - 2

: integrate

- x^{2} + 2

⇥1.3) Let's compute the integral:

\int_{- \sqrt{2}}^{\sqrt{2}} (- x^{2} + 2) d x

⇥ⅰ

\int (- x^{2} + 2) d x

- \int x^{2} d x + \int 2 d x

⇥

- \int x^{2} d x + \int 2 d x

⇥ⓘ An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

- \int x^{2} d x

- \frac{x^{3}}{3}

■The derivative of

2 x

2

, so

2 x

is an antiderivative of

2

\int 2 d x

2 x

▷

- \frac{x^{3}}{3} + 2 x

⇥1.4) Let's evaluate for the interval

[- \sqrt{2}, \sqrt{2}]

(2 \cdot \sqrt{2} - \frac{{(\sqrt{2})}^{3}}{3}) - (2 \cdot (- \sqrt{2}) - \frac{{(- \sqrt{2})}^{3}}{3})

⇥ⓘ A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

{(\sqrt{2})}^{3}

2^{\frac{1}{2} \cdot 3}

■A minus sign before a parenthesis changes the sign of each of its terms:

- (2 \cdot (- \sqrt{2}) - \frac{{(- \sqrt{2})}^{3}}{3})

- 2 \cdot (- \sqrt{2}) + \frac{{(- \sqrt{2})}^{3}}{3}

⇥

2 \cdot \sqrt{2}

- \frac{2^{\frac{1}{2} \cdot 3}}{3}

- 2 \cdot (- \sqrt{2})

+ \frac{{(- \sqrt{2})}^{3}}{3}

⇥ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{1}{2} \cdot 3

\frac{1 \cdot 3}{2}

\frac{3}{2}

■Two minus signs cancel out:

-2 \cdot (- \sqrt{2})

2 \cdot \sqrt{2}

■To add fractions, put them over the same denominator, then add the numerators:

- \frac{2^{\frac{3}{2}}}{3} + \frac{{(- \sqrt{2})}^{3}}{3}

\frac{- 2^{\frac{3}{2}} + {(- \sqrt{2})}^{3}}{3}

⇥

2 \cdot \sqrt{2} + \frac{- 2^{\frac{3}{2}} + {(- \sqrt{2})}^{3}}{3} + 2 \cdot \sqrt{2}

⇥ⓘ Without parentheses, the minus sign is not part of the power:

- 2^{\frac{3}{2}}

- 2 \cdot \sqrt{2}

■Like terms are grouped by adding their coefficients:

2 \cdot \sqrt{2} + 2 \cdot \sqrt{2}

4 \cdot \sqrt{2}

■Powers are computed before multiplications, additions and subtractions. With an even integer exponent, the minus sign in the base disappears; with an odd integer exponent, it moves in front of the power:

{(- \sqrt{2})}^{3}

- {(\sqrt{2})}^{3}

⇥

4 \cdot \sqrt{2} + \frac{- 2 \cdot \sqrt{2} - {(\sqrt{2})}^{3}}{3}

⇥ⓘ The minus sign is factored out, changing the sign of each term:

- 2 \cdot \sqrt{2} - {(\sqrt{2})}^{3}

- (2 \cdot \sqrt{2} + {(\sqrt{2})}^{3})

⇥

4 \cdot \sqrt{2} - \frac{2 \cdot \sqrt{2} + {(\sqrt{2})}^{3}}{3}

⇥ⅰ

4 \cdot \sqrt{2} - \frac{2 \cdot \sqrt{2} + {(\sqrt{2})}^{3}}{3}

\frac{3 \cdot 4 \cdot \sqrt{2} - (2 \cdot \sqrt{2} + {(\sqrt{2})}^{3})}{3}

⇥

\frac{3 \cdot 4 \cdot \sqrt{2} - (2 \cdot \sqrt{2} + {(\sqrt{2})}^{3})}{3}

⇥ⓘ Multiplications are done before additions and subtractions:

3 \cdot 4

12

■A minus sign before a parenthesis changes the sign of each of its terms:

- (2 \cdot \sqrt{2} + {(\sqrt{2})}^{3})

- 2 \cdot \sqrt{2} - {(\sqrt{2})}^{3}

⇥

\frac{12 \cdot \sqrt{2} - 2 \cdot \sqrt{2} - {(\sqrt{2})}^{3}}{3}

⇥ⓘ Like terms are grouped by adding their coefficients:

12 \cdot \sqrt{2} - 2 \cdot \sqrt{2}

10 \cdot \sqrt{2}

■A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

{(\sqrt{2})}^{3}

2^{\frac{1}{2} \cdot 3}

⇥

\frac{10 \cdot \sqrt{2} - 2^{\frac{1}{2} \cdot 3}}{3}

⇥ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{1}{2} \cdot 3

\frac{1 \cdot 3}{2}

\frac{3}{2}

■What the terms have in common is factored out, so that the fraction can be simplified:

\frac{10 \cdot \sqrt{2} - 2^{\frac{3}{2}}}{3}

\frac{2 \cdot \sqrt{2} \cdot (5 - 1)}{3}

⇥

\frac{2 \cdot \sqrt{2} \cdot (5 - 1)}{3}

⇥ⓘ Parentheses are computed first:

5 - 1

4

⇥

\frac{2 \cdot \sqrt{2} \cdot 4}{3}

⇥ⅰ

2 \cdot 4

8

⇥

\frac{8 \cdot \sqrt{2}}{3}

The area on

[- \sqrt{2}, \sqrt{2}]

\frac{8 \cdot \sqrt{2}}{3}

[\sqrt{2}, 2]

x^{2} - 2

is above

0

: integrate

x^{2} - 2

⇥1.5) Let's compute the integral:

\int_{\sqrt{2}}^{2} (x^{2} - 2) d x

⇥ⅰ

\int (x^{2} - 2) d x

\int x^{2} d x - \int 2 d x

⇥

\int x^{2} d x - \int 2 d x

⇥ⓘ An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int x^{2} d x

\frac{x^{3}}{3}

■The derivative of

- 2 x

-2

, so

- 2 x

is an antiderivative of

-2

- \int 2 d x

- 2 x

▷

\frac{x^{3}}{3} - 2 x

⇥1.6) Let's evaluate over the interval

[\sqrt{2}, 2]

(\frac{2^{3}}{3} - 2 \cdot 2) - (\frac{{(\sqrt{2})}^{3}}{3} - 2 \cdot \sqrt{2})

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{3}

2 \cdot 2 \cdot 2

8

■Multiplications are done before additions and subtractions:

2 \cdot 2

4

■A minus sign before a parenthesis changes the sign of each of its terms:

- (\frac{{(\sqrt{2})}^{3}}{3} - 2 \cdot \sqrt{2})

- \frac{{(\sqrt{2})}^{3}}{3} + 2 \cdot \sqrt{2}

⇥

\frac{8}{3}

- 4

- \frac{{(\sqrt{2})}^{3}}{3}

+ 2 \cdot \sqrt{2}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{8}{3} - 4

\frac{8 - 4 \cdot 3}{3}

■A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

{(\sqrt{2})}^{3}

2^{\frac{1}{2} \cdot 3}

⇥

\frac{8 - 4 \cdot 3}{3} - \frac{2^{\frac{1}{2} \cdot 3}}{3} + 2 \cdot \sqrt{2}

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 3

12

■Multiplying a fraction by a number multiplies its numerator:

\frac{1}{2} \cdot 3

\frac{1 \cdot 3}{2}

\frac{3}{2}

⇥

\frac{8 - 12}{3} - \frac{2^{\frac{3}{2}}}{3} + 2 \cdot \sqrt{2}

⇥ⓘ The numerator and the denominator are computed before the division:

8 - 12

-4

■Powers are computed before multiplications, additions and subtractions:

2^{\frac{3}{2}}

2 \cdot \sqrt{2}

⇥

- \frac{4}{3} - \frac{2 \cdot \sqrt{2}}{3} + 2 \cdot \sqrt{2}

⇥ⓘ Like terms are grouped by adding their coefficients:

- \frac{2 \cdot \sqrt{2}}{3} + 2 \cdot \sqrt{2}

\frac{4 \cdot \sqrt{2}}{3}

⇥

- \frac{4}{3} + \frac{4 \cdot \sqrt{2}}{3}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators:

- \frac{4}{3} + \frac{4 \cdot \sqrt{2}}{3}

\frac{- 4 + 4 \cdot \sqrt{2}}{3}

⇥

\frac{- 4 + 4 \cdot \sqrt{2}}{3}

⇥

\frac{4 \cdot \sqrt{2} - 4}{3}

The area on

[\sqrt{2}, 2]

\frac{4 \cdot \sqrt{2} - 4}{3}

⇥

\frac{4 \cdot \sqrt{2} - 4}{3} + \frac{8 \cdot \sqrt{2}}{3} + \frac{4 \cdot \sqrt{2} - 4}{3}

⇥ⓘ Like terms are grouped by adding their coefficients:

\frac{4 \cdot \sqrt{2} - 4}{3} + \frac{4 \cdot \sqrt{2} - 4}{3}

\frac{2 \cdot (4 \cdot \sqrt{2} - 4)}{3}

⇥

\frac{2 \cdot (4 \cdot \sqrt{2} - 4)}{3} + \frac{8 \cdot \sqrt{2}}{3}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{2 \cdot (4 \cdot \sqrt{2} - 4)}{3} + \frac{8 \cdot \sqrt{2}}{3}

\frac{2 \cdot (4 \cdot \sqrt{2} - 4) + 8 \cdot \sqrt{2}}{3}

⇥

\frac{2 \cdot (4 \cdot \sqrt{2} - 4) + 8 \cdot \sqrt{2}}{3}

⇥ⓘ Expand by multiplying each term inside the parentheses:

2 \cdot (4 \cdot \sqrt{2} - 4)

8 \cdot \sqrt{2} - 8

⇥

\frac{8 \cdot \sqrt{2} - 8 + 8 \cdot \sqrt{2}}{3}

ⓘ Like terms are grouped by adding their coefficients:

8 \cdot \sqrt{2} + 8 \cdot \sqrt{2}

16 \cdot \sqrt{2}

▶Area:

\frac{16 \cdot \sqrt{2} - 8}{3}

2) The numeric value of:

\frac{16 \cdot \sqrt{2} - 8}{3}

equals:

\frac{16 \cdot \approx 1.4142 - 8}{3}

\frac{\approx 22.6274 - 8}{3}

\frac{\approx 14.6274}{3}

▷

\approx 4.8758