QUESTION 8 The volume of the solid obtained by rotating the region bounded by $$ y=10-6 x+x^{2} $$ and $$ y=-10+6 x-x^{2} $$ and $$ x=1 $$ and $$ x=5 $$ about $$ y=8 $$ is

Question

UpStudy AI · Accepted Answer

To find the volume of the solid obtained by rotating the region bounded by the curves $$ y = 10 - 6x + x^2 $$ and $$ y = -10 + 6x - x^2 $$ around the line $$ y = 8 $$, we can use the method of cylindrical shells or the washer method. Here, we'll use the washer method.

### Step 1: Identify the curves and the region of integration

The curves are:
1. $$ y_1 = 10 - 6x + x^2 $$
2. $$ y_2 = -10 + 6x - x^2 $$

We need to find the points of intersection of these two curves to determine the limits of integration.

### Step 2: Set the equations equal to find points of intersection

Set $$ y_1 = y_2 $$:
$$
10 - 6x + x^2 = -10 + 6x - x^2
$$

### Step 3: Rearranging the equation

Rearranging gives:
$$
2x^2 - 12x + 20 = 0
$$

### Step 4: Solve the quadratic equation

We can simplify this equation:
$$
x^2 - 6x + 10 = 0
$$

Now, we will use the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$:
- $$ a = 1 $$
- $$ b = -6 $$
- $$ c = 10 $$

Calculating the discriminant:
$$
b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot 10 = 36 - 40 = -4
$$

Since the discriminant is negative, the curves do not intersect, meaning they are always separated in the given range.

### Step 5: Determine the volume using the washer method

The volume $$ V $$ of the solid of revolution is given by:
$$
V = \pi \int_{a}^{b} \left( R^2 - r^2 \right) \, dx
$$
where $$ R $$ is the outer radius and $$ r $$ is the inner radius.

In this case:
- The outer radius $$ R = 8 - y_1 $$
- The inner radius $$ r = 8 - y_2 $$

### Step 6: Set up the integral

The limits of integration are from $$ x = 1 $$ to $$ x = 5 $$:
$$
V = \pi \int_{1}^{5} \left( (8 - (10 - 6x + x^2))^2 - (8 - (-10 + 6x - x^2))^2 \right) \, dx
$$

This simplifies to:
$$
V = \pi \int_{1}^{5} \left( (6x - x^2 - 2)^2 - (6x - x^2 + 18)^2 \right) \, dx
$$

### Step 7: Calculate the integral

Now, we will calculate the integral:
$$
V = \pi \int_{1}^{5} \left( (6x - x^2 - 2)^2 - (6x - x^2 + 18)^2 \right) \, dx
$$

Let's compute this integral.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\pi \times \int_{1}^{5} \left(\left(6x-x^{2}-2\right)^{2}-\left(6x-x^{2}+18\right)^{2}\right) dx$$
- step1: Subtract the terms:
  $$\pi \times \int_{1}^{5} \left(40x^{2}-240x-320\right) dx$$
- step2: Evaluate the integral:
  $$\pi \left(-\frac{7520}{3}\right)$$
- step3: Multiply the numbers:
  $$-\pi \times \frac{7520}{3}$$
- step4: Multiply:
  $$-\frac{\pi \times 7520}{3}$$
- step5: Multiply:
  $$-\frac{7520\pi }{3}$$
The volume $$ V $$ of the solid obtained by rotating the region bounded by the curves $$ y = 10 - 6x + x^2 $$ and $$ y = -10 + 6x - x^2 $$ around the line $$ y = 8 $$ is:

$$
V = -\frac{7520\pi}{3}
$$

However, since volume cannot be negative, we take the absolute value:

$$
V = \frac{7520\pi}{3}
$$

Thus, the final answer for the volume is:

$$
V = \frac{7520\pi}{3}
$$
The volume of the solid is $$ \frac{7520\pi}{3} $$.

QUESTION 8 The volume of the solid obtained by rotating the region bounded by \( y=10-6 x+x^{2} \) and \( y=-10+6 x-x^{2} \) and \( x=1 \) and \( x=5 \) about \( y=8 \) is

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions