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

1) Consider the two curves with equations

y = x^{2}

and

y = 2 x

At the intersections,

x^{2}

2 x

. Subtract the second expression from the first:

x^{2} - 2 x = 0

⇥1.1) Let's solve in

ℝ

x^{2} - 2 x = 0

⇥You can factorize the expression:

x^{2} - 2 x

into:

x \cdot (x - 2)

⇥A product is zero if and only if one of its factors is zero: we solve each factor equal to 0

⇥⇥1.1.1) Let's solve in

ℝ

x = 0

⇥⇥Solution obtained:

x = 0

⇥⇥1.1.2) Let's solve in

ℝ

x - 2 = 0

⇥⇥Solution obtained:

x = 2

The area between

x^{2}

and

2 x

[0, 2]

is the integral of the absolute value of their difference

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

[0, 2]

2 x

is above

x^{2}

: integrate

- x^{2} + 2 x

⇥1.2) Let's integrate:

\int_{0}^{2} (- x^{2} + 2 x) d x

⇥ⅰ

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

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

⇥

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

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

\int 2 x d x

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

⇥

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

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

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

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

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

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

x^{2}

▷

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

⇥1.3) Let's evaluate over the interval

[0, 2]

(2^{2} - \frac{2^{3}}{3}) - (0^{2} - \frac{0^{3}}{3})

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

2^{2}

2 \cdot 2

4

2^{3}

2 \cdot 2 \cdot 2

8

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

- (0^{2} - \frac{0^{3}}{3})

- 0^{2} + \frac{0^{3}}{3}

⇥

4

- \frac{8}{3}

- 0^{2}

+ \frac{0^{3}}{3}

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

4 - \frac{8}{3}

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

■Adding 0 changes nothing:

0 + 0

0

⇥

\frac{3 \cdot 4 - 8}{3} + 0

⇥ⓘ Adding 0 changes nothing:

\frac{3 \cdot 4 - 8}{3} + 0

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

■Multiplications are done before additions and subtractions:

3 \cdot 4

12

⇥

\frac{12 - 8}{3}

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

12 - 8

4

⇥

\frac{4}{3}

The area on

[0, 2]

\frac{4}{3}

▶Area:

\frac{4}{3}

2) The numeric value of:

\frac{4}{3}

is:

▷

\approx 1.3333