Math Snap

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.

The approximate volume of the solid is 12.0192 cubic units.