Volumes with Cross Sections: Triangles and Semicircles

Volumes with Cross Sections: Triangles and Semicircles - AP Calculus AB/BC Study Guide 2024

Hello numbers wizards and calculus conjurers! Are you prepared to dive into the mathematics of 3D shapes like a calculus superhero? 🦸‍♂️🦸‍♀️ Today, we’re turning abstract x-y graphs into concrete (well, maybe not concrete, but definitely three-dimensional) solids! We'll dissect these volumes, using cross sections that are triangles and semicircles. Ready, set, integrate! 📏🏔️

To find the volume of a solid with known cross sections, we employ the grand wizardry formula of integration: [ V = \int_a^b A(x) , dx ] Here, (A(x)) is the area of a cross section (which could be a mystical triangle or a magical semicircle) perpendicular to the ( x )-axis over the interval ([a, b]). By dividing solids into ultra-thin slices, akin to slicing bread but with a lot more math, we can simplify our calculations.

Triangles come in different flavors, just like ice cream! The area formula for a triangle cross section depends on its type.

Equilateral Triangles

Let's start with the perfect, balanced equilateral triangle 🛤️. An equilateral triangle has all three sides of equal length (s). The area formula for such a triangle is: [ A(x) = \frac{\sqrt{3}}{4} s^2 ] Thus, the volume of a solid with equilateral triangle cross sections is: [ V = \int_a^b \frac{\sqrt{3}}{4} s^2 , dx ]

Right Isosceles Triangles

Next up is the right-angled isosceles triangle ⛰️. Think of it as the vanilla triangle with a 90-degree twist! For this triangle, where the two equal sides are (s), the area formula is: [ A(x) = \frac{1}{2} s^2 ] Hence, the volume of a solid with right-angled isosceles triangle cross sections is given by: [ V = \int_a^b \frac{1}{2} s^2 , dx ]

Semicircles are like half-pizzas (and who doesn't love pizza?)! 🍕 To find the area of any semicircle, we use the formula: [ A(x) = \frac{1}{2} \pi r^2 ] where (r) denotes the radius of the semicircle. Therefore, the volume of a solid with semicircular cross sections is: [ V = \int_a^b \frac{1}{2} \pi r^2 , dx ]

Picture this. You have a region bounded by (y = x^2) and ( y = \sqrt{x} ). Imagine you decided to turn this bounded area into a 3D solid with cross sections perpendicular to the ( x )-axis. What integral would give you the volume if the cross sections are equilateral triangles? And how about semicircles?

Step 1: Visualize and Boundaries

First, draw the graphs of ( y = \sqrt{x} ) and ( y = x^2 ). Your graphs might intersect at ( x = 0 ) and ( x = 1 ), setting our integration bounds as the interval ([0, 1]).

Step 2: Find the Side Length or Radius

For each cross section:

• Equilateral Triangle: At any point ( x ) between 0 and 1, the side length ( s ) of the equilateral triangle is given by the difference between the curves: ( s = \sqrt{x} - x^2 ).

• Semicircle: Similarly, the diameter of the semicircle ( D ) is ( \sqrt{x} - x^2 ), and thus the radius ( r ) is half of that: ( r = \frac{\sqrt{x} - x^2}{2} ).

Step 3: Set up the Area Function

For the equilateral triangle: [ A(x) = \frac{\sqrt{3}}{4} (\sqrt{x} - x^2)^2 ]

For the semicircle: [ A(x) = \frac{1}{2} \pi \left(\frac{\sqrt{x} - x^2}{2}\right)^2 = \frac{\pi}{8} (\sqrt{x} - x^2)^2 ]

Step 4: Build the Volume Integral

Plugging in our functions, we get:

Equilateral Triangle: [ V = \frac{\sqrt{3}}{4} \int_0^1 (\sqrt{x} - x^2)^2 , dx ]

Semicircle: [ V = \frac{\pi}{8} \int_0^1 (\sqrt{x} - x^2)^2 , dx ]

Step 5: Solve the Integral

Now, integrate these functions. I promise solving is worth it—like unwrapping a calculus gift! 🎁📚

Equilateral Triangle Integral Solution: [ \frac{\sqrt{3}}{4} \int_0^1 (\sqrt{x} - x^2)^2 , dx \approx \boxed{\frac{3\sqrt{3}}{280}} ]

Semicircle Integral Solution: [ \frac{\pi}{8} \int_0^1 (\sqrt{x} - x^2)^2 , dx \approx \boxed{\frac{9\pi}{560}} ]

Whenever we aim to find the volume of a solid with known cross sections, our go-to equation is:

[ V=∫_{a}^{b}A(x) , dx ]

Key Area Functions:

• Equilateral triangles: ( A(x) = \frac{\sqrt{3}}{4} s^2 )
• Right isosceles triangles: ( A(x) = \frac{1}{2} s^2 )
• Semicircles: ( A(x) = \frac{1}{2} \pi r^2 )

To solve these problems, determine ( s ) or ( r ), determine the bounds, set up the integral, and evaluate!

• Axis of Revolution: Imaginary line around which a shape rotates to create a 3D object.
• Cross Sections: The 2D shapes obtained when a solid is cut by a plane.
• Definite Integral: Mathematical tool for calculating exact area or volume over an interval.
• Radius: The distance from the center to the circumference of a circle.
• Semicircular Cross Sections: Using semicircles as the base to find volume.

So there we have it! You’re now equipped to tackle these integrals with the confidence of a math guru and the excitement of uncovering the volumes that lie within your curves. Go forth and integrate, young Jedi of calculus! 🚀📐