Question

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

Short answer

The area of the region is\(\frac{{\rm{1}}}{{\rm{5}}}{\rm{units}}\).

Step-by-step solution

What’s Given

Given \({\rm{y = }}{{\rm{x}}^{\frac{{\rm{1}}}{{\rm{4}}}}}{\rm{,x = }}{{\rm{y}}^{\rm{4}}}\)

We have: right boundary of the region is the\({\rm{x = }}{{\rm{y}}^{\rm{4}}}\)and the left boundary is the y-axis (line\({\rm{x = 0}}\)) within the region, the y values vary from\({\rm{y = 0 to y = 1}}\).

Area of the region.

\(\begin{aligned}{l}\int {{\rm{(rigth - left)dy}}} \\{\rm{from,}}\\{\rm{y &= 0 to y &= 1}}\end{aligned}\)

\(\begin{aligned}{l}{\rm{ &= }}\int {{\rm{(}}{{\rm{y}}^{\rm{4}}}{\rm{ - 0)dy }}} {\rm{from,y &= 0 to y &= 1}}\\{\rm{ &= }}\left( {\frac{{{{\rm{y}}^{\rm{5}}}}}{{\rm{5}}}} \right)_{\rm{0}}^{\rm{1}}\\{\rm{ &= }}\frac{{\rm{1}}}{{\rm{5}}}{{\rm{(1)}}^{\rm{5}}}{\rm{ - }}\frac{{\rm{1}}}{{\rm{5}}}{{\rm{(0)}}^{\rm{5}}}\\{\rm{ &= }}\frac{{\rm{1}}}{{\rm{5}}}{\rm{ units}}{\rm{.}}\end{aligned}\)