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

1) Consider the two curves with equations

x = y^{2}

and

x = 2 y

At the intersections,

y^{2}

2 y

. Subtract the second expression from the first:

y^{2} - 2 y = 0

⇥1.1) Let's solve in

ℝ

y^{2} - 2 y = 0

⇥You can factorize the expression:

y^{2} - 2 y

into:

y \cdot (y - 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

y = 0

⇥⇥Solution obtained:

y = 0

⇥⇥1.1.2) Let's solve in

ℝ

y - 2 = 0

⇥⇥Solution found:

y = 2

The area between

y^{2}

and

2 y

[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 y

is to the right of

y^{2}

: integrate

- y^{2} + 2 y

with respect to

y

⇥1.2) Let's compute the integral:

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

⇥ⅰ

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

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

⇥

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

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

- \int y^{2} d y

- \frac{y^{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 y d y

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

⇥

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

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

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

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

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

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

y^{2}

▷

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

⇥1.3) Let's evaluate for 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}

equals:

▷

\approx 1.3333