Find the area of the region bounded by the graphs of the given equations. y = 3, y = (square root...

Question:

Find the area of the region bounded by the graphs of the given equations. {eq}y = 3, \: y = \sqrt[4]{x}, \: x = 0. {/eq}

Finding the Area Between Two Curves:

To find the area bounded by the graphs of {eq}x = 0, \: y = c, \: y = f(x), {/eq} where {eq}f(x) \leq c, {/eq} we first calculate the intersection point where {eq}f(x) = c. {/eq} Call the solution {eq}x_1. {/eq} If this value is greater than zero, then the area between the two graphs is

{eq}A = \displaystyle\int_0^{x_1} c - f(x) \: dx. {/eq}

Answer and Explanation: 1

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The intersection of these two curves occurs when {eq}\sqrt[4]{x} = 3, {/eq} so {eq}x = 81. {/eq} Therefore the area between the curves is equal to

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