Question

The boundaries of the shaded region are the y-axis, the line y = 15, and the curve y = 15 4 x . Find the area of this region by writing x as a function of y and integrating with respect to y.

Answer 1

Answer:

Area = 3

Step-by-step explanation:

In given figure curve is given as:

                               $y=15\sqrt[4]{x}$

                               $\frac{y}{15}=\sqrt[4]{x}$

                               $(\frac{y}{15})^{4}=x$

                               $x=\frac{y^{4}}{15^{4}}$ ---(1)

Area of bounded region is found by integrating above equation (1) between limits 0 to 15.

                 $A=\frac{1}{15^{4}}\int\limits^{15}_{0} {y^{4}} \, dy$

                 $A=\frac{1}{15^{4}}[\frac{y^{5}}{5}]^{15}_{0}$

                 $A=\frac{1}{15^{4}}[\frac{15^{5}}{5}-0]$

                 $A=\frac{15}{5}$

                 $A=3$

Answer 2

Answer:

Area of the region is 3

Step-by-step explanation:

In given figure curve is given as:

                     $y=15\sqrt[4]{x}$

Dividing 15 both the sides, we get

                     $\frac{y}{15} =\frac{15}{15} \sqrt[4]{x}$

     or             $\frac{y}{15} =\sqrt[4]{x}$

     or            $(\frac{y}{15}) ^{4} =x$

     or            $x=\frac{y^{4} }{15^{4} }$   ........(i)

Area of bounded region is found by integrating above equation (i) between 0 to 15.

                   $A=\frac{1}{15^{4} } \int\limits^a_b {y^{4} } \, dy$

Here: a = 0 and b = 15

                $A=\frac{1}{15^{4} } (\frac{y^{5} }{5} )\\A=\frac{1}{15^{4} }(\frac{15^{5} }{5}-0)\\A=\frac{15}{5} \\A=3$

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