Math / Calculus

QuestionUse the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x$ -axis. (Round your answer to four decimal places.)
$y = 2 \arctan(0.4x)$ $y = 0$ $x = 0$ $x = 3$

Studdy Solution

STEP 1

What is this asking? We need to find the volume of a funky shape made by spinning a curvy region around the x-axis! Watch out! Remember to use radians in your calculator when working with $\arctan$ and be super careful with your parentheses when setting up the integral.

STEP 2

1. Set up the integral
2. Evaluate the integral

STEP 3

Imagine the area trapped between the curve $y = 2 \arctan(0.4x)$ , the x-axis ( $y=0$ ), the y-axis ( $x=0$ ), and the vertical line $x=3$ .
Now, imagine spinning this area around the x-axis.
It'll create a cool 3D shape, kind of like a vase!

STEP 4

We'll use the disk method to find the volume.
Each "disk" is a thin slice of our 3D shape, perpendicular to the x-axis.
The area of each disk is $\pi \cdot (\text{radius})^2$ .
Since our curve $y = 2 \arctan(0.4x)$ gives us the radius at any $x$ value, the area of each disk is $\pi \cdot (2 \arctan(0.4x))^2$ .

STEP 5

To find the total volume, we'll add up the volumes of all these super thin disks using an integral.
The limits of integration are from $x=0$ to $x=3$ , the boundaries of our region.
So, our integral is: $V = \int_{0}^{3} \pi (2 \arctan(0.4x))^2 \, dx$ $V = \pi \int_{0}^{3} 4 (\arctan(0.4x))^2 \, dx$ $V = 4\pi \int_{0}^{3} (\arctan(0.4x))^2 \, dx$

STEP 6

This integral is tricky to solve by hand, so we'll use a graphing utility (like a graphing calculator).
Make sure it's set to radians!

STEP 7

Carefully enter the integral into your calculator: $4\pi \int_{0}^{3} (\arctan(0.4x))^2 \, dx$ You should get approximately 12.0192.

STEP 8

The approximate volume of the solid is 12.0192 cubic units.

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QuestionUse the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x$ -axis. (Round your answer to four decimal places.)
$y = 2 \arctan(0.4x)$ $y = 0$ $x = 0$ $x = 3$

Studdy Solution

STEP 1

What is this asking? We need to find the volume of a funky shape made by spinning a curvy region around the x-axis! Watch out! Remember to use radians in your calculator when working with $\arctan$ and be super careful with your parentheses when setting up the integral.

STEP 2

1. Set up the integral
2. Evaluate the integral

STEP 3

Imagine the area trapped between the curve $y = 2 \arctan(0.4x)$ , the x-axis ( $y=0$ ), the y-axis ( $x=0$ ), and the vertical line $x=3$ .
Now, imagine spinning this area around the x-axis.
It'll create a cool 3D shape, kind of like a vase!

STEP 4

STEP 5

STEP 6

This integral is tricky to solve by hand, so we'll use a graphing utility (like a graphing calculator).
Make sure it's set to radians!

STEP 7

Carefully enter the integral into your calculator: $4\pi \int_{0}^{3} (\arctan(0.4x))^2 \, dx$ You should get approximately 12.0192.

STEP 8

The approximate volume of the solid is 12.0192 cubic units.

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Studdy Solution

STEP 1

What is this asking? We need to find the volume of a funky shape made by spinning a curvy region around the x-axis! Watch out! Remember to use radians in your calculator when working with arctan⁡\arctanarctan and be super careful with your parentheses when setting up the integral.

STEP 2

1. Set up the integral2. Evaluate the integral

STEP 3

Imagine the area trapped between the curve y=2arctan⁡(0.4x)y = 2 \arctan(0.4x)y=2arctan(0.4x), the x-axis (y=0y=0y=0), the y-axis (x=0x=0x=0), and the vertical line x=3x=3x=3.Now, imagine spinning this area around the x-axis.It'll create a cool 3D shape, kind of like a vase!

STEP 4

STEP 5

STEP 6

This integral is tricky to solve by hand, so we'll use a graphing utility (like a graphing calculator).Make sure it's set to radians!

STEP 7

Carefully enter the integral into your calculator: 4π∫03(arctan⁡(0.4x))2 dx 4\pi \int_{0}^{3} (\arctan(0.4x))^2 \, dx 4π∫03​(arctan(0.4x))2dx You should get approximately **12.0192**.

STEP 8

The approximate volume of the solid is **12.0192** cubic units.

Get Studdy

Studdy solves anything!

Start learning now

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

What is this asking? We need to find the volume of a funky shape made by spinning a curvy region around the x-axis! Watch out! Remember to use radians in your calculator when working with $\arctan$ and be super careful with your parentheses when setting up the integral.

1. Set up the integral
2. Evaluate the integral

Imagine the area trapped between the curve $y = 2 \arctan(0.4x)$ , the x-axis ( $y=0$ ), the y-axis ( $x=0$ ), and the vertical line $x=3$ .
Now, imagine spinning this area around the x-axis.
It'll create a cool 3D shape, kind of like a vase!

This integral is tricky to solve by hand, so we'll use a graphing utility (like a graphing calculator).
Make sure it's set to radians!

Carefully enter the integral into your calculator: $4\pi \int_{0}^{3} (\arctan(0.4x))^2 \, dx$ You should get approximately 12.0192.

The approximate volume of the solid is 12.0192 cubic units.