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LammettHash
[tex]y=25x^2\implies x=\pm5\sqrt y[/tex]

The area between the parabola and the line is

[tex]\displaystyle\int_0^4(5\sqrt y-(-5\sqrt y))\,\mathrm dy=10\int_0^4\sqrt y\,\mathrm dy=\dfrac{10}{\frac32}y^{3/2}\bigg|_{y=0}^{y=4}=\dfrac{20}3\times8=\frac{160}3[/tex]

But you're looking for the line [tex]y=b[/tex] such that the areas of each subdivision are equal. This means you want [tex]b[/tex] to satisfy

[tex]\displaystyle10\int_0^b\sqrt y\,\mathrm dy=\dfrac{80}3[/tex]

You have

[tex]\displaystyle\int_0^b\sqrt y\,\mathrm dy=\dfrac83[/tex]
[tex]\dfrac23y^{3/2}\bigg|_{y=0}^{y=b}=\dfrac83[/tex]
[tex]\dfrac23b^{3/2}=\dfrac83[/tex]
[tex]b^{3/2}=4[/tex]
[tex]b=4^{2/3}=2\sqrt[3]2\approx2.52[/tex]
vijayhalder031

The correct answer is [tex]2 \sqrt[3]{2}[/tex] ≈ 2.52

What is a Parabola?

  • A symmetrical plane curve that forms when a cone intersects with a plane parallel to its side.

How to Solve the Problem?

The problem can be solved by following steps.

[tex]y = 25x^2[/tex] (Given)

y = 4 into two regions with equal area (Given)

We need to find b in the equation

So, The area of region given by

[tex]\int\limits^2_ {-2} (4-x^{2} ) \, dx \\= 2\int\limits^2_0 (4-x^{2} )\, dx\\[/tex]

= [tex]2(4x-\frac{x^{3} }{3})^0_2[/tex]

=[tex]\frac{32}{3}[/tex]

Second, y=b intersects the curve

x=±[tex]\sqrt{b}[/tex]

We want to find b such that

[tex]\int\limits^a_b {(b-x^2)} \, dx[/tex]

Whereas , a = [tex]\sqrt{b}[/tex] and b = [tex]-\sqrt{b}[/tex]

= [tex]\frac{16}{3}[/tex]

This will only occur if and only if

[tex]\int\limits^a_b {(b-x^2)} \, dx[/tex]  Where the limits are a =[tex]\sqrt{b}[/tex] and b =0

= 8/3

Integrating it we will get

[tex]{(bx - \frac{x^3}{3}) }\limits^a_b[/tex] = 8/3

Where a = [tex]\sqrt{b}[/tex], b = 0

[tex]b\sqrt{b}-\frac{b\sqrt{b}}{3}\\[/tex] = 8/3

Therefore,  [tex]\frac{2b\sqrt{b} }{3}[/tex] = 8/3

[tex]b\sqrt{b} =4[/tex]

[tex]b^{\frac{3}{2}} = 4[/tex]

[tex]b=\sqrt[3]{16}[/tex]

[tex]b=2\sqrt[3]{2}[/tex]

Round answer in two decimal places ≈2.52

Hence the number b ≈ 2.52

Learn more about Integration here

https://brainly.in/question/3259157

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