Find solutions to your problems with the help of IDNLearn.com's expert community. Ask your questions and get detailed, reliable answers from our community of knowledgeable experts.
Sagot :
Certainly! Let's find the volume formed by rotating the region enclosed by the curves [tex]\(x = 2.5y\)[/tex] and [tex]\(x = y^3\)[/tex] with [tex]\(y \geq 0\)[/tex] about the [tex]\(y\)[/tex]-axis.
### Step 1: Find the Points of Intersection
First, we need to determine where the two curves intersect. To find these points, we set the equations equal to each other:
[tex]\[ 2.5y = y^3 \][/tex]
Rearranging the equation, we get:
[tex]\[ y^3 - 2.5y = 0 \][/tex]
Factor out [tex]\(y\)[/tex]:
[tex]\[ y(y^2 - 2.5) = 0 \][/tex]
This gives us the solutions:
[tex]\[ y = 0 \quad \text{or} \quad y^2 = 2.5 \][/tex]
Solving for [tex]\(y^2 = 2.5\)[/tex], we get:
[tex]\[ y = \sqrt{2.5} \quad \text{or} \quad y = -\sqrt{2.5} \][/tex]
However, since we are only considering [tex]\(y \geq 0\)[/tex]:
[tex]\[ y = 0 \quad \text{and} \quad y = \sqrt{2.5} \][/tex]
Note that [tex]\(\sqrt{2.5}\)[/tex] can be simplified further to:
[tex]\[ y = \sqrt{2.5} = \sqrt{2.5} = \sqrt{ \frac{25}{10}} = \frac{5}{\sqrt{10}} = \frac{5 \cdot \sqrt{10}}{10} = \frac{\sqrt{25}}{\sqrt{4}} = 1.58113883008419 \][/tex]
Thus, the points of intersection are:
[tex]\[ y = 0 \quad \text{and} \quad y = 1.58113883008419 \][/tex]
### Step 2: Set Up the Integral for the Washer Method
Next, we rotate the region around the [tex]\(y\)[/tex]-axis. This requires us to employ the washer method to find the volume. The volume formula for the washer method when rotating about the [tex]\(y\)[/tex]-axis is given by:
[tex]\[ V = \pi \int_{a}^{b} \left( R(y)^2 - r(y)^2 \right) dy \][/tex]
Here, [tex]\(R(y)\)[/tex] is the outer radius, and [tex]\(r(y)\)[/tex] is the inner radius for the given [tex]\(y\)[/tex]. In our case:
- The outer radius is given by the line [tex]\(x = 2.5y\)[/tex].
- The inner radius is given by the curve [tex]\(x = y^3\)[/tex].
So, [tex]\(R(y) = 2.5y\)[/tex] and [tex]\(r(y) = y^3\)[/tex]. Therefore, the volume integral becomes:
[tex]\[ V = \pi \int_{0}^{1.58113883008419} \left[ (2.5y)^2 - (y^3)^2 \right] dy \][/tex]
Simplify the integrand:
[tex]\[ (2.5y)^2 = 6.25y^2 \][/tex]
[tex]\[ (y^3)^2 = y^6 \][/tex]
Thus, the integral is:
[tex]\[ V = \pi \int_{0}^{1.58113883008419} (6.25y^2 - y^6) \, dy \][/tex]
### Step 3: Compute the Integral
Now, integrate the function within the bounds:
[tex]\[ \int_{0}^{1.58113883008419} (6.25y^2 - y^6) \, dy \][/tex]
Find the indefinite integral first:
[tex]\[ \int (6.25y^2 - y^6) \, dy = 6.25 \int y^2 \, dy - \int y^6 \, dy \][/tex]
This gives:
[tex]\[ 6.25 \cdot \frac{y^3}{3} - \frac{y^7}{7} = \frac{6.25}{3}y^3 - \frac{y^7}{7} \][/tex]
Now, we evaluate this expression from 0 to 1.58113883008419:
[tex]\[ \left[ \frac{6.25}{3} y^3 - \frac{y^7}{7} \right]_{0}^{1.58113883008419} \][/tex]
At [tex]\(y = 1.58113883008419\)[/tex]:
[tex]\[ \frac{6.25}{3} (1.58113883008419)^3 - \frac{(1.58113883008419)^7}{7} = 4.70577032763152 \][/tex]
At [tex]\(y = 0\)[/tex], the terms vanish. So:
[tex]\[ \left[ \frac{6.25}{3} y^3 - \frac{y^7}{7} \right]_{0}^{1.58113883008419} = 4.70577032763152 \][/tex]
### Step 4: Calculate the Volume
Finally, multiply by [tex]\(\pi\)[/tex]:
[tex]\[ V = \pi \times 4.70577032763152 = 14.7836134907680 \][/tex]
Therefore, the volume formed by rotating the given region about the [tex]\(y\)[/tex]-axis is:
[tex]\[ \boxed{14.7836134907680} \][/tex]
### Step 1: Find the Points of Intersection
First, we need to determine where the two curves intersect. To find these points, we set the equations equal to each other:
[tex]\[ 2.5y = y^3 \][/tex]
Rearranging the equation, we get:
[tex]\[ y^3 - 2.5y = 0 \][/tex]
Factor out [tex]\(y\)[/tex]:
[tex]\[ y(y^2 - 2.5) = 0 \][/tex]
This gives us the solutions:
[tex]\[ y = 0 \quad \text{or} \quad y^2 = 2.5 \][/tex]
Solving for [tex]\(y^2 = 2.5\)[/tex], we get:
[tex]\[ y = \sqrt{2.5} \quad \text{or} \quad y = -\sqrt{2.5} \][/tex]
However, since we are only considering [tex]\(y \geq 0\)[/tex]:
[tex]\[ y = 0 \quad \text{and} \quad y = \sqrt{2.5} \][/tex]
Note that [tex]\(\sqrt{2.5}\)[/tex] can be simplified further to:
[tex]\[ y = \sqrt{2.5} = \sqrt{2.5} = \sqrt{ \frac{25}{10}} = \frac{5}{\sqrt{10}} = \frac{5 \cdot \sqrt{10}}{10} = \frac{\sqrt{25}}{\sqrt{4}} = 1.58113883008419 \][/tex]
Thus, the points of intersection are:
[tex]\[ y = 0 \quad \text{and} \quad y = 1.58113883008419 \][/tex]
### Step 2: Set Up the Integral for the Washer Method
Next, we rotate the region around the [tex]\(y\)[/tex]-axis. This requires us to employ the washer method to find the volume. The volume formula for the washer method when rotating about the [tex]\(y\)[/tex]-axis is given by:
[tex]\[ V = \pi \int_{a}^{b} \left( R(y)^2 - r(y)^2 \right) dy \][/tex]
Here, [tex]\(R(y)\)[/tex] is the outer radius, and [tex]\(r(y)\)[/tex] is the inner radius for the given [tex]\(y\)[/tex]. In our case:
- The outer radius is given by the line [tex]\(x = 2.5y\)[/tex].
- The inner radius is given by the curve [tex]\(x = y^3\)[/tex].
So, [tex]\(R(y) = 2.5y\)[/tex] and [tex]\(r(y) = y^3\)[/tex]. Therefore, the volume integral becomes:
[tex]\[ V = \pi \int_{0}^{1.58113883008419} \left[ (2.5y)^2 - (y^3)^2 \right] dy \][/tex]
Simplify the integrand:
[tex]\[ (2.5y)^2 = 6.25y^2 \][/tex]
[tex]\[ (y^3)^2 = y^6 \][/tex]
Thus, the integral is:
[tex]\[ V = \pi \int_{0}^{1.58113883008419} (6.25y^2 - y^6) \, dy \][/tex]
### Step 3: Compute the Integral
Now, integrate the function within the bounds:
[tex]\[ \int_{0}^{1.58113883008419} (6.25y^2 - y^6) \, dy \][/tex]
Find the indefinite integral first:
[tex]\[ \int (6.25y^2 - y^6) \, dy = 6.25 \int y^2 \, dy - \int y^6 \, dy \][/tex]
This gives:
[tex]\[ 6.25 \cdot \frac{y^3}{3} - \frac{y^7}{7} = \frac{6.25}{3}y^3 - \frac{y^7}{7} \][/tex]
Now, we evaluate this expression from 0 to 1.58113883008419:
[tex]\[ \left[ \frac{6.25}{3} y^3 - \frac{y^7}{7} \right]_{0}^{1.58113883008419} \][/tex]
At [tex]\(y = 1.58113883008419\)[/tex]:
[tex]\[ \frac{6.25}{3} (1.58113883008419)^3 - \frac{(1.58113883008419)^7}{7} = 4.70577032763152 \][/tex]
At [tex]\(y = 0\)[/tex], the terms vanish. So:
[tex]\[ \left[ \frac{6.25}{3} y^3 - \frac{y^7}{7} \right]_{0}^{1.58113883008419} = 4.70577032763152 \][/tex]
### Step 4: Calculate the Volume
Finally, multiply by [tex]\(\pi\)[/tex]:
[tex]\[ V = \pi \times 4.70577032763152 = 14.7836134907680 \][/tex]
Therefore, the volume formed by rotating the given region about the [tex]\(y\)[/tex]-axis is:
[tex]\[ \boxed{14.7836134907680} \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.
