integral from -1 to 2 of (y+2-y^2) — step-by-step solution

1) Let's compute the integral:

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

ⅰ

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

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

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

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

\int y d y

\frac{y^{2}}{2}

- \int y^{2} d y

- \frac{y^{3}}{3}

■The derivative of

2 y

2

, so

2 y

is an antiderivative of

2

\int 2 d y

2 y

▷

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

2) Let's evaluate over 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}

▶Integral:

\frac{9}{2}

3) The numeric value of:

\frac{9}{2}

is:

▷

4.5