### Mastering Volumes with Cross Sections: Squares and Rectangles in AP Calculus AB/BC

#### Introduction

Hey calculus enthusiasts! Ready to turn those flat two-dimensional curves into jaw-dropping three-dimensional masterpieces? Hold onto your graphing calculators, because we're venturing into the wonderful world of finding volumes with cross sections. We're talking about slicing and dicing shapes like a pro chef, but mathematically. 🧮✂️

#### Concept of Cross Sections

Imagine you're making a multi-layer cake (yum!). Each layer is thin—a cross-section of the entire cake. To find the volume of the whole cake, we consider the volume of these very thin slices. Similarly, in calculus, we can break down complex three-dimensional solids into infinitely thin slices to calculate their total volume. The magical formula we use for this is: [ V = \int_a^b A(x) , dx ] Here, ( A(x) ) represents the area of a cross-section at ( x ), and ( dx ) is the infinitesimally small thickness of each slice.

#### Square Cross Sections

For a solid with square cross sections perpendicular to the x-axis, the area ( A(x) ) of each square slice is simply ( s^2 ), where ( s ) is the length of a side of the square. Plugging this into our volume formula gives: [ V = \int_a^b s^2 , dx ] It's like stacking up an infinite number of paper-thin square slices to build our three-dimensional shape.

#### Rectangular Cross Sections

When dealing with rectangular cross sections, the area ( A(x) ) is found using ( w \times h ), where ( w ) is the width and ( h ) is the height of the rectangle. The formula for the volume then becomes: [ V = \int_a^b w \times h , dx ] Think of it as piling up rectangles to form a solid, like building with Jenga blocks, but smoother. 😉

#### Problem-Solving with Square Cross Sections

Let’s tackle a classic example to flex those calculus muscles!

**Example 1: Solids with Square Cross Sections**

Consider a region bounded by ( y = x^2 ) and ( y = \sqrt{x} ). Each cross section perpendicular to the x-axis is a square. We need to find the volume of this square-shaped monster.

Taking ( h(x) = \sqrt{x} ) and ( g(x) = x^2 ) as our upper and lower bounds, respectively, the side length ( s ) of the square cross section at any ( x ) is the difference between these two functions, ( s = \sqrt{x} - x^2 ).

The bounds of integration are determined by the points of intersection of ( y = x^2 ) and ( y = \sqrt{x} ). Setting ( \sqrt{x} = x^2 ) and solving, we get ( x = 0 ) and ( x = 1 ).

So, the volume ( V ) is: [ V = \int_0^1 (\sqrt{x} - x^2)^2 , dx ]

After all the fancy math jitters, solving this integral results in: [ V = \frac{9}{70} \approx 0.1285 ] Voilà! We've turned our flat area into a solid volume.

#### Problem-Solving with Rectangular Cross Sections

**Example 2: Solids with Rectangular Cross Sections**

Let’s consider a solid whose base is defined by ( y = x^3 ), ( y = 0 ), and ( x = 2 ). The cross sections, taken perpendicular to the y-axis, form rectangles with a constant height of 6.

Firstly, we rewrite ( y = x^3 ) in terms of ( x ): ( x = \sqrt[3]{y} ).

The width ( w ) at any ( y ) is ( 2 - \sqrt[3]{y} ), since one boundary is ( x = 2 ). With a constant height ( h ) of 6, the volume ( V ) formula is: [ V = \int_0^8 (2 - \sqrt[3]{y}) \times 6 , dy ]

Solving the integral, we get: [ V = 24 ] And just like that, we've created a rectangular volume!

#### Recap Time!

Remember, finding the volume of a solid with known cross sections is like solving a delicious puzzle:

- Identify the area ( A(x) ) of the cross section.
- Determine the bounds of integration.
- Plug everything into the integral and solve. Easy-peasy, right?

For square cross sections, the area is ( A(x) = s^2 ). For rectangular cross sections, it's ( A(x) = w \times h ).

So next time you're asked to calculate these volumes, you’ll be cool as a cucumber, slicing and dicing those integrals like a math ninja! 🥷📐

Happy Calculating! 🚀