integral from 3.5 to 7 of 274000-39200x

1) Let's integrate:

\int_{3.5}^{7} (274,000 - 39,200 x) d x

ⅰ

\int (274,000 - 39,200 x) d x

\int 274,000 d x - \int 39,200 x d x

\int 274,000 d x - \int 39,200 x d x

ⓘ The derivative of

274,000 x

274,000

, so

274,000 x

is an antiderivative of

274,000

\int 274,000 d x

274,000 x

■A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

- \int 39,200 x d x

- 39,200 \cdot \frac{x^{2}}{2}

274,000 x - 39,200 \cdot \frac{x^{2}}{2}

ⓘ Multiplying a fraction by a number multiplies its numerator:

39,200 \cdot \frac{x^{2}}{2}

\frac{39,200 x^{2}}{2}

■A fraction is simplified by dividing the numerator and the denominator by the same number:

- \frac{39,200 x^{2}}{2}

- 19,600 x^{2}

274,000 x - 19,600 x^{2}

You can write the expression in factored form:

x \cdot (274,000 - 19,600 x)

Let's factor in

ℝ

(274,000 - 19,600 x)

We factor out

400

, the greatest common divisor of the coefficients:

274,000 - 19,600 x

400 \cdot (685 - 49 x)

Factor found:

685 - 49 x

▷

400 x \cdot (685 - 49 x)

2) Let's evaluate over the interval

[3.5, 7]

400 \cdot 7 \cdot (685 - 49 \cdot 7) - 400 \cdot 3.5 \cdot (685 - 49 \cdot 3.5)

ⓘ Multiplications are done before additions and subtractions:

49 \cdot 7

(40 + 9) \cdot 7

40 \cdot 7 + 9 \cdot 7

280 + 63

343

400 \cdot 7

2800

■Multiplications are done before additions and subtractions. Multiply without the decimal points, then put the point back by counting the decimals of both factors:

49 \cdot 3.5

\frac{49 \cdot 35}{10}

\frac{1715}{10}

171.5

400 \cdot 3.5

\frac{400 \cdot 35}{10}

\frac{14,000}{10}

1400

2800 \cdot (685 - 343) - 1400 \cdot (685 - 171.5)

ⓘ Parentheses are computed first:

685 - 343

342

685 - 171.5

513.5

2800 \cdot 342 - 1400 \cdot 513.5

ⓘ Multiplications are done before additions and subtractions:

2800 \cdot 342

2800 \cdot (300 + 40 + 2)

2800 \cdot 300 + 2800 \cdot 40 + 2800 \cdot 2

840,000 + 112,000 + 5600

957,600

■Multiplications are done before additions and subtractions. Multiply without the decimal points, then put the point back by counting the decimals of both factors:

1400 \cdot 513.5

\frac{1400 \cdot 5135}{10}

\frac{7,189,000}{10}

718,900

957,600 - 718,900

ⓘ Start from

957,600

and move back

718,900

on the number line:

957,600 - 718,900

238,700

▶Integral:

238,700