## Task Condition

Calculate the area of the figure bounded by the graphs of the functions:

$$
y=(x-1)^{2}, y^{2}=x-1
$$
## Solution

$$
y=(x-1)^{2}, y^{2}=x-1
$$

## Integrals 14-29

The function $y^{2}=x-1$ can be represented as two functions $y$ of $x$:

$$
\begin{aligned}
& y=\sqrt{x-1} \\
& y=-\sqrt{x-1}
\end{aligned}
$$

Based on the graphs of the functions, the required figure is bounded only by the graphs $y=(x-1)^{2}, y=\sqrt{x-1}$. Let's find the limits of integration:

$$
\begin{aligned}
& y=(x-1)^{2}=\left(y^{2}\right)^{2} \\
& y^{4}-y=0 \\
& y\left(y^{3}-1\right)=0 \\
& y(y-1)\left(y^{2}+y+1\right)=0 \\
& y=0 \text { or } y=1 \\
& y^{2}+y+1>0 \text { since } \\
& D=1^{2}-4 \cdot 1 \cdot 1=-3
\end{aligned}
$$

Then:

$$
\begin{aligned}
& y=0 \Rightarrow x=1 \\
& y=1 \Rightarrow x=2
\end{aligned}
$$

Therefore, the limits of integration are:

![](https://cdn.mathpix.com/cropped/2024_05_22_aa6951b1e92815ea34b6g-55.jpg?height=1388&width=962&top_left_y=1019&top_left_x=981)
1 and 2.

Let's find the area of the figure as the difference between the areas under the graphs $y=\sqrt{x-1}$ and $y=(x-1)^{2}$

$$
\int_{1}^{2}(\sqrt{x-1}) d x-\int_{1}^{2}(x-1)^{2} d x=
$$

$$
\begin{aligned}
& =\int_{1}^{2}(x-1)^{\frac{1}{2}} d x-\int_{1}^{2}\left(x^{2}-2 x+1\right) d x= \\
& =\left.\frac{2}{3}(x-1)^{\frac{3}{2}}\right|_{1} ^{2}-\left.\left(\frac{x^{3}}{3}-\frac{2 x^{2}}{2}+x\right)\right|_{1} ^{2}= \\
& =\frac{2}{3}-\frac{8}{3}+\frac{8}{2}-2+\frac{1}{3}-1+1 \\
& =\frac{2}{3}-\frac{8}{3}+\frac{8}{2}-2+\frac{1}{3}-1+1=\frac{1}{3}
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+14-29$ »

Categories: Kuznetsov's Problem Book Integrals Problem 14 | Integrals

Ukrainian Banner Network

- Last modified: 22:34, 9 June 2010.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 14-30

## Material from PlusPi