integral from 2 to 3 of (x^2-4) — step-by-step solution

1) Let's compute the integral:

\int_{2}^{3} (x^{2} - 4) d x

ⅰ

\int (x^{2} - 4) d x

\int x^{2} d x - \int 4 d x

\int x^{2} d x - \int 4 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

- 4 x

-4

, so

- 4 x

is an antiderivative of

-4

- \int 4 d x

- 4 x

▷

\frac{x^{3}}{3} - 4 x

2) Let's evaluate over the interval

[2, 3]

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

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

3^{3}

3 \cdot 3 \cdot 3

27

■Multiplications are done before additions and subtractions:

4 \cdot 3

12

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

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

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

\frac{27}{3}

- 12

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

+ 4 \cdot 2

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

3

, their greatest common divisor:

\frac{27}{3}

9

■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:

4 \cdot 2

8

9

- 12

- \frac{8}{3}

+ 8

ⓘ Start from

9

and move back

12

on the number line, going below zero:

9 - 12

-3

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

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

- 3 - \frac{8}{3}

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

\frac{3 \cdot (-3) - 8}{3} + 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:

3 \cdot (-3)

-9

\frac{- 9 - 8}{3} + 8

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

- 9 - 8

-17

- \frac{17}{3} + 8

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

- \frac{17}{3} + 8

\frac{- 17 + 8 \cdot 3}{3}

\frac{- 17 + 8 \cdot 3}{3}

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 3

24

\frac{- 17 + 24}{3}

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

- 17 + 24

7

▶Integral:

\frac{7}{3}

3) The numeric value of:

\frac{7}{3}

equals:

▷

\approx 2.3333