area 3x,7x^2 — step-by-step solution

⚠Ⓘ 3x is read as the product 3*x: for a power, write 3^x‖7x is read as the product 7*x: for a power, write 7^x

y

is already isolated in

y = 3 x

y

is already isolated in

y = 7 x^{2}

At the intersections,

3 x

7 x^{2}

. Subtract the second expression from the first:

3 x - 7 x^{2} = 0

⇥You can write the expression:

3 x - 7 x^{2}

in factored form:

x \cdot (3 - 7 x)

⇥Solution found:

x = 0

⇥1.1) Let's solve in

ℝ

3 - 7 x = 0

⇥

7 x = 3

⇥Solution found:

x = \frac{3}{7}

The area between

3 x

and

7 x^{2}

[0, \frac{3}{7}]

is the integral of the absolute value of their difference

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

[0, \frac{3}{7}]

3 x

is above

7 x^{2}

: integrate

3 x - 7 x^{2}

⇥1.2) Let's compute the integral:

\int(3 x - 7 x^{2})dx

⇥

\int(3 x)dx - \int(7 x^{2})dx

⇥ℹ

\int(3 x)dx

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

‖

- \int(7 x^{2})dx

- 7 \cdot \frac{x^{3}}{3}

⇥

3 \cdot \frac{x^{2}}{2} - 7 \cdot \frac{x^{3}}{3}

⇥ℹ

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

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

‖

- 7 \cdot \frac{x^{3}}{3}

- \frac{7 x^{3}}{3}

▷

\frac{3 x^{2}}{2} - \frac{7 x^{3}}{3}

⇥1.3) Let's evaluate over the interval

[0, \frac{3}{7}]

(- \frac{7 \cdot {(\frac{3}{7})}^{3}}{3} + \frac{3 \cdot {(\frac{3}{7})}^{2}}{2}) - (- \frac{7 \cdot 0^{3}}{3} + \frac{3 \cdot 0^{2}}{2})

⇥

- \frac{7 \cdot {(\frac{3}{7})}^{3}}{3}

+ \frac{3 \cdot {(\frac{3}{7})}^{2}}{2}

+ \frac{7 \cdot 0^{3}}{3}

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

⇥ℹ

7 \cdot 0

0

‖

3 \cdot 0

0

⇥

- \frac{7 \cdot {(\frac{3}{7})}^{3}}{3}

+ \frac{3 \cdot {(\frac{3}{7})}^{2}}{2}

+ 0

+ 0

⇥

- \frac{7 \cdot {(\frac{3}{7})}^{3}}{3} + \frac{3 \cdot {(\frac{3}{7})}^{2}}{2}

⇥ℹ

{(\frac{3}{7})}^{3}

\frac{3^{3}}{7^{3}}

‖

{(\frac{3}{7})}^{2}

\frac{3^{2}}{7^{2}}

⇥

- \frac{7 \cdot \frac{3^{3}}{7^{3}}}{3} + \frac{3 \cdot \frac{3^{2}}{7^{2}}}{2}

⇥ℹ

3^{3}

27

‖

7^{3}

343

‖

3^{2}

9

‖

7^{2}

49

⇥

- \frac{7 \cdot \frac{27}{343}}{3} + \frac{3 \cdot \frac{9}{49}}{2}

⇥ℹ

7 \cdot \frac{27}{343}

\frac{189}{343}

‖

3 \cdot \frac{9}{49}

\frac{27}{49}

⇥

- \frac{\frac{189}{343}}{3} + \frac{\frac{27}{49}}{2}

⇥

- \frac{27}{49} \cdot \frac{1}{3} + \frac{27}{49} \cdot \frac{1}{2}

⇥ℹ

- \frac{27}{49} \cdot \frac{1}{3}

- \frac{27}{147}

‖

\frac{27}{49} \cdot \frac{1}{2}

\frac{27}{98}

⇥

- \frac{27}{147} + \frac{27}{98}

⇥

\frac{- 27 \cdot 2 + 27 \cdot 3}{147 \cdot 2}

⇥ℹ

-27 \cdot 2

-54

‖

27 \cdot 3

81

‖

147 \cdot 2

294

⇥

\frac{- 54 + 81}{294}

⇥

\frac{27}{294}

⇥

\frac{9}{98}

The area on

[0, \frac{3}{7}]

\frac{9}{98}

▶Area:

\frac{9}{98}

2) The numeric value of:

\frac{9}{98}

equals:

▷

\approx 9.1837 \times 10^{- 2}