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.


Referring to the graph:

enter image description here

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

  • $\begingroup$ Good. But Let me ask you one question. The lower part of figure (under the line $y=1$) is also bounded by curve and two lines right? $\endgroup$ – ZFR Sep 27 '17 at 6:58
  • $\begingroup$ No, if below $y=1$ is considered, then the figure will be bounded by the curve on the left, by the line $y=1$ above and by the line $x=8$ on the right, however, no bound from below. $\endgroup$ – farruhota Sep 27 '17 at 7:03
  • $\begingroup$ So, the only region bounded by all three functions is the shaded region indicated on the graph above. $\endgroup$ – farruhota Sep 27 '17 at 7:04
  • $\begingroup$ It will be bounded from below by $x$-axis right? Though there is nothing about that, right? $\endgroup$ – ZFR Sep 27 '17 at 7:06
  • $\begingroup$ It depends on with respect to $x$ or $y$. If w.r.t. $x$, then bounded by $y=x^{1/3}$ from above and $y=1$ from below. If w.r.t. $y$, then bounded by $x=8$ from above and $x=y^3$ from below. So, look at the $\Delta x$ and $\Delta y$. $\endgroup$ – farruhota Sep 27 '17 at 7:47

I think you didn't understand well what area you should calculate. You should calculate the area above the $y=1$ line and hence it would be:

$$\int_1^2 (8 - y^3)dy = 8y - \frac {y^4}{4}\bigg|_1^8 = 16 - 8 -\frac{16}{4} + \frac 14 = 4 + \frac 14$$

You need to find the area of the shaded region.

  • $\begingroup$ I have drawn a picture but I can't see anything false. Could you provide picture please? $\endgroup$ – ZFR Sep 27 '17 at 6:55
  • $\begingroup$ @RFZ I've added one in the post $\endgroup$ – Stefan4024 Sep 27 '17 at 7:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.