Calculate the volume of the solid generated by revolving the region bounded by the curve $$ y = x^2 $$ and the line $$ y = 4 $$ around the x-axis.

Question

UpStudy AI · Accepted Answer

To calculate the volume of the solid generated by revolving the region bounded by the curve $$ y = x^2 $$ and the line $$ y = 4 $$ around the x-axis, we'll use the **washer method**. This method is ideal for finding volumes of solids of revolution when the region is bounded between two curves.

### Step 1: Determine the Points of Intersection
First, find where the curves $$ y = x^2 $$ and $$ y = 4 $$ intersect:
$$
x^2 = 4 \implies x = \pm 2
$$
So, the region of interest is between $$ x = -2 $$ and $$ x = 2 $$.

### Step 2: Set Up the Washer Method Integral
When revolving around the x-axis, the volume $$ V $$ can be calculated using:
$$
V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 \right] dx
$$
where:
- $$ R(x) $$ is the outer radius (distance from the x-axis to the outer curve),
- $$ r(x) $$ is the inner radius (distance from the x-axis to the inner curve).

In this case:
- The outer curve is $$ y = 4 $$, so $$ R(x) = 4 $$.
- The inner curve is $$ y = x^2 $$, so $$ r(x) = x^2 $$.

Thus, the integral becomes:
$$
V = \pi \int_{-2}^{2} \left[ 4^2 - (x^2)^2 \right] dx = \pi \int_{-2}^{2} \left[ 16 - x^4 \right] dx
$$

### Step 3: Evaluate the Integral
Notice that the integrand $$ 16 - x^4 $$ is an even function, so we can simplify the computation by integrating from 0 to 2 and doubling the result:
$$
V = 2\pi \int_{0}^{2} \left[ 16 - x^4 \right] dx
$$
$$
= 2\pi \left[ 16x - \frac{x^5}{5} \right]_{0}^{2}
$$
$$
= 2\pi \left[ 16(2) - \frac{2^5}{5} \right]
$$
$$
= 2\pi \left[ 32 - \frac{32}{5} \right]
$$
$$
= 2\pi \left[ \frac{160}{5} - \frac{32}{5} \right]
$$
$$
= 2\pi \left[ \frac{128}{5} \right]
$$
$$
= \frac{256\pi}{5}
$$

### Final Answer
The volume of the solid is $$ \dfrac{256}{5} $$ multiplied by π. Thus,

$$
V = \frac{256}{5}\, \pi
$$
The volume of the solid is $$ \frac{256}{5} \pi $$.

Calculate the volume of the solid generated by revolving the region bounded by the curve and the line around the x-axis.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Calculus Questions