<s>[INST] The area of curve $y^3=x$ bounded by lines. Find the area of region on $xy$ plane shaded by curve $y^3=x$ and lines $y=1$ and $x=8$
My solution: The line $x=8$ intersect curve at point $y=2$ and line $y=1$ intersect at point x=1. So the intended area is $$=\left(1-\int \limits_{0}^{1}y^3dy\right)+7=8-\frac{1}{4}=7,75$$
However, the answer in the book is $4,25$.
Can anyone point out where is my mistake with graphs? I have rechecked many times and did not detected it.
 [/INST] Referring to the graph:

1-method:
$$S=\int_{x=1}^{x=8} x^{1/3}dx-7\cdot 1=\left(\frac{3}{4} x^{4/3}\right) \bigg|_1^8-7=\left(12-\frac34\right)-7=4\frac14.$$
2-method:
$$S=8\cdot 1-\int_{y=1}^{y=2} y^3dy=8-\left(\frac{1}{4} y^4\right) \bigg|_1^2=8-\left(4-\frac14\right)=4\frac14.$$
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