area y=x^2,y=x+2 — step-by-step solution

1) Consider the two curves with equations

y = x^{2}

and

y = x + 2

At the intersections,

x^{2}

x + 2

. Subtract the second expression from the first:

x^{2} - (x + 2)

= 0, giving

x^{2} - x - 2 = 0

⇥1.1) Let's solve in

ℝ

x^{2} - x - 2 = 0

⇥Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -1

and

c = -2

⇥

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-2)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

⇥

Δ = 1 - 4 \cdot (-2)

⇥ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-2)

8

⇥

Δ = 1 + 8

⇥ⓘ Start from

1

and move forward

8

on the number line:

1 + 8

9

⇥

Δ = 9

⇥As

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{9}}{2 \cdot 1}

⇥

x_{1} = \frac{1 - \sqrt{9}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{9}

3

;

3^{2}

9

⇥

x_{1} = \frac{1 - 3}{2}

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

1 - 3

-2

⇥

x_{1} = - \frac{2}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{2}{2}

-1

⇥

x_{1} = -1

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{9}}{2 \cdot 1}

⇥

x_{2} = \frac{1 + \sqrt{9}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{9}

3

;

3^{2}

9

⇥

x_{2} = \frac{1 + 3}{2}

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

1 + 3

4

⇥

x_{2} = \frac{4}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{4}{2}

2

⇥

x_{2} = 2

⇥Solutions found:

x = {-1, 2}

The area between

x^{2}

and

x + 2

[-1, 2]

is the integral of the absolute value of their difference

No interior intersection: the difference keeps its sign on the interval

[-1, 2]

x + 2

is above

x^{2}

: integrate

- x^{2} + x + 2

⇥1.2) Let's integrate:

\int_{-1}^{2} (- x^{2} + x + 2) d x

⇥ⅰ

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

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

⇥

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

\int x d x

\frac{x^{2}}{2}

■The derivative of

2 x

2

, so

2 x

is an antiderivative of

2

\int 2 d x

2 x

▷

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

⇥1.3) Let's evaluate for the interval

[-1, 2]

(- \frac{2^{3}}{3} + \frac{2^{2}}{2} + 2 \cdot 2) - (- \frac{{(-1)}^{3}}{3} + \frac{{(-1)}^{2}}{2} + 2 \cdot (-1))

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

2^{3}

2 \cdot 2 \cdot 2

8

2^{2}

2 \cdot 2

4

■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{{(-1)}^{3}}{3} + \frac{{(-1)}^{2}}{2} + 2 \cdot (-1))

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

⇥

- \frac{8}{3}

+ \frac{4}{2}

+ 4

+ \frac{{(-1)}^{3}}{3}

- \frac{{(-1)}^{2}}{2}

- 2 \cdot (-1)

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators. We choose

6

, the least common multiple of

3

and

2

. Multiplying the numerator and the denominator by the same number does not change the fraction:

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

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

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

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

{(-1)}^{3}

(-1) \cdot (-1) \cdot (-1)

-1

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-2 \cdot (-1)

2

⇥

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

+ 4

- \frac{1}{3}

- \frac{1}{2}

+ 2

⇥ⓘ Multiplications are done before additions and subtractions:

8 \cdot 2

16

4 \cdot 3

12

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

4 - \frac{1}{3}

\frac{3 \cdot 4 - 1}{3}

⇥ⅰ

3 \cdot 2

6

⇥

\frac{- 16 + 12}{6}

+ \frac{3 \cdot 4 - 1}{3}

- \frac{1}{2}

+ 2

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

- 16 + 12

-4

■Multiplications are done before additions and subtractions:

3 \cdot 4

12

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

- \frac{1}{2} + 2

\frac{- 1 + 2 \cdot 2}{2}

⇥

- \frac{4}{6} + \frac{12 - 1}{3} + \frac{- 1 + 2 \cdot 2}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{4}{6}

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

- \frac{2}{3}

■The numerator and the denominator are computed before the division:

12 - 1

11

■Multiplications are done before additions and subtractions:

2 \cdot 2

4

⇥

- \frac{2}{3} + \frac{11}{3} + \frac{- 1 + 4}{2}

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

- \frac{2}{3} + \frac{11}{3}

\frac{- 2 + 11}{3}

■The numerator and the denominator are computed before the division:

- 1 + 4

3

⇥

\frac{- 2 + 11}{3} + \frac{3}{2}

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

- 2 + 11

9

⇥

\frac{9}{3} + \frac{3}{2}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators. We choose

6

, the least common multiple of

3

and

2

. Multiplying the numerator and the denominator by the same number does not change the fraction:

\frac{9}{3} + \frac{3}{2}

\frac{9 \cdot 2}{3 \cdot 2} + \frac{3 \cdot 3}{2 \cdot 3}

\frac{9 \cdot 2 + 3 \cdot 3}{3 \cdot 2}

⇥

\frac{9 \cdot 2 + 3 \cdot 3}{3 \cdot 2}

⇥ⓘ Multiplications are done before additions and subtractions:

9 \cdot 2

18

3 \cdot 3

9

⇥ⅰ

3 \cdot 2

6

⇥

\frac{18 + 9}{6}

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

18 + 9

27

⇥

\frac{27}{6}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

3

, their greatest common divisor:

\frac{27}{6}

\frac{9 \cdot 3}{2 \cdot 3}

\frac{9}{2}

⇥

\frac{9}{2}

The area on

[-1, 2]

\frac{9}{2}

▶Area:

\frac{9}{2}

2) The numeric value of:

\frac{9}{2}

is:

▷

4.5