Find the region's area that is enclosed by the curves $$ y = x^{3} $$ and $$ y = 3x - x^{2} $$.

Question

UpStudy AI · Accepted Answer

To find the area enclosed by the curves $$ y = x^3 $$ and $$ y = 3x - x^2 $$, follow these steps:

### 1. Find the Points of Intersection
Set the two equations equal to each other to find the points where the curves intersect:

$$
x^3 = 3x - x^2
$$

Rearrange the equation:

$$
x^3 + x^2 - 3x = 0 \
x(x^2 + x - 3) = 0
$$

So, the roots are:
$$
x = 0
$$
$$
x = \frac{-1 \pm \sqrt{13}}{2}
$$

Therefore, the points of intersection are:
$$
x = \frac{-1 - \sqrt{13}}{2}, \quad x = 0, \quad x = \frac{-1 + \sqrt{13}}{2}
$$

### 2. Determine Which Function is on Top
To determine which function is above the other between the points of intersection, pick test points in each interval.

- **Between $$ \frac{-1 - \sqrt{13}}{2} $$ and $$ 0 $$:**
  
  Choose $$ x = -1 $$:
  $$
  y = (-1)^3 = -1 \
  y = 3(-1) - (-1)^2 = -4
  $$
  Here, $$ y = x^3 > y = 3x - x^2 $$.

- **Between $$ 0 $$ and $$ \frac{-1 + \sqrt{13}}{2} $$:**
  
  Choose $$ x = 1 $$:
  $$
  y = 1^3 = 1 \
  y = 3(1) - 1^2 = 2
  $$
  Here, $$ y = 3x - x^2 > y = x^3 $$.

### 3. Set Up the Integral
The total area is the sum of the areas between the curves in each interval:

$$
\text{Area} = \int_{\frac{-1 - \sqrt{13}}{2}}^{0} \left( x^3 - (3x - x^2) \right) dx + \int_{0}^{\frac{-1 + \sqrt{13}}{2}} \left( (3x - x^2) - x^3 \right) dx
$$

### 4. Calculate the Integral
After simplifying and evaluating the integrals, we find:

$$
\text{Area} = \frac{73}{12}
$$

### Final Answer
The area enclosed by the curves is $$ \boxed{\dfrac{73}{12}} $$.
The area enclosed by the curves $$ y = x^3 $$ and $$ y = 3x - x^2 $$ is $$ \frac{73}{12} $$.

Find the region’s area that is enclosed by the curves and .

Solución de inteligencia artificial de Upstudy

Responder

Solución

1. Find the Points of Intersection

2. Determine Which Function is on Top

3. Set Up the Integral

4. Calculate the Integral

Final Answer

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions