Question

Find the area of the region bounded by the graphs of the given functions y=x⁴−4x² y=4x² Give your numerical answer (approximate number only) in square units to three decimal places of precision.

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Frederik

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64 Answers

To find the area of the region bounded by the graphs of the functions y = x^4 - 4x^2 and y = 4x^2 , we first need to find the points of intersection:

Setting the two equations equal to each other, we have:

x^4 - 4x^2 = 4x^2

x^4 - 8x^2 = 0

x^2(x^2 - 8) = 0

This gives usx = 0, \sqrt{8}, -\sqrt{8} as points of intersection.

The area of the region bounded by the two functions is given by:

A = 2 \times \int_{0}^{\sqrt{8}} |4x^2 - (x^4 - 4x^2)| dx

Simplify the absolute value expression inside the integral:

A = 2 \times \int_{0}^{\sqrt{8}} |4x^2 - x^4 + 4x^2| dx

A = 2 \times \int_{0}^{\sqrt{8}} |8x^2 - x^4| dx

Further simplifying:

A = 2 \times \int_{0}^{\sqrt{8}} (x^4 - 8x^2) dx

Integrating term by term:

A = 2 \times (\frac{x^5}{5} - \frac{8x^3}{3}) \Big|_{0}^{\sqrt{8}}

A = 2 \times (\frac{8\sqrt{8}}{5} - \frac{8 \cdot 8\sqrt{8}}{3})

A = \frac{16\sqrt{8}}{5} - \frac{128\sqrt{8}}{3}

A = \frac{48\sqrt{8}}{15}

Therefore, the area of the region bounded by the graphs of the functionsy = x^4 - 4x^2 and y = 4x^2 is \frac{48\sqrt{8}}{15} square units.

\boxed{A = \frac{48\sqrt{8}}{15}}=48.272

Setting the two equations equal to each other, we have:

This gives us

The area of the region bounded by the two functions is given by:

Simplify the absolute value expression inside the integral:

Further simplifying:

Integrating term by term:

Therefore, the area of the region bounded by the graphs of the functions

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