Math  /  Calculus

QuestionPeriod: DUE: Tuesdav, 11/19/24
1. Find the volume of the solid obtained by rotating the region given about the x -axis and using the given bounds. Sketch a picture of the graph. y=0.3x3y=0.3 x^{3}, the xx- axis, and x=3x=3

Studdy Solution

STEP 1

What is this asking? We need to find out how much space is inside a 3D shape created by spinning the curve y=0.3x3 y = 0.3x^3 from x=0 x = 0 to x=3 x = 3 around the x-axis. Watch out! Don't forget to use the right formula for volume when rotating around the x-axis.
And make sure to set the limits of integration correctly!

STEP 2

1. Set up the integral
2. Calculate the integral
3. Find the volume

STEP 3

Alright, let's start by **defining the function** we're working with: y=0.3x3 y = 0.3x^3 .
We're going to rotate this curve around the x-axis, which means we're creating a solid of revolution.
To find the volume of this solid, we use the disk method!

STEP 4

The formula for the volume V V of a solid of revolution using the disk method is:
V=πab[f(x)]2dxV = \pi \int_{a}^{b} [f(x)]^2 \, dx
Here, f(x)=0.3x3 f(x) = 0.3x^3 , and our bounds are from x=0 x = 0 to x=3 x = 3 .

STEP 5

Let's **plug the function into the formula**.
We have:
V=π03(0.3x3)2dxV = \pi \int_{0}^{3} (0.3x^3)^2 \, dx

STEP 6

Now, let's **simplify the expression** inside the integral:
(0.3x3)2=0.09x6(0.3x^3)^2 = 0.09x^6

STEP 7

So, our integral becomes:
V=π030.09x6dxV = \pi \int_{0}^{3} 0.09x^6 \, dx

STEP 8

Let's **integrate** 0.09x6 0.09x^6 :
0.09x6dx=0.09x77=0.097x7\int 0.09x^6 \, dx = 0.09 \cdot \frac{x^7}{7} = \frac{0.09}{7} x^7

STEP 9

Now, let's **evaluate this from 0 to 3**:
V=π[0.097(3)70.097(0)7]V = \pi \left[ \frac{0.09}{7} (3)^7 - \frac{0.09}{7} (0)^7 \right]

STEP 10

Calculate 37 3^7 :
37=21873^7 = 2187

STEP 11

Substitute back into the expression:
V=π[0.0972187]V = \pi \left[ \frac{0.09}{7} \cdot 2187 \right]

STEP 12

Calculate 0.0921877 \frac{0.09 \cdot 2187}{7} :
0.0921877=28.11\frac{0.09 \cdot 2187}{7} = 28.11

STEP 13

Finally, multiply by π\pi:
V=28.11πV = 28.11\pi

STEP 14

The volume of the solid obtained by rotating the region around the x-axis is approximately 28.11π 28.11\pi cubic units.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord