Volume with Washer Method: Revolving Around Other Axes - AP Calculus AB/BC Study Guide 2024
Introduction
Hello, math enthusiasts and intrepid integral explorers! 🚀 Welcome to the grand finale of Unit 8 in AP Calculus AB, where we tackle the volume using the Washer Method, but with a fun twist—we'll revolve around axes that aren't the ol' boring x- or y-axes. Grab your calculators, and let's spin some nerdy magic!
Revolving Around Other Axes 🌪️
When we say "revolving around other axes," it means we're spicing things up by rotating our region around lines like ( x = a ) or ( y = b ) rather than just sticking to the x- or y-axis. It’s like asking your hamster to run on a wheel of a different color—changes the perspective but the workout is the same.
The generic formula for revolving around a horizontal line ( y = b ) is: [ \int_{c}^{d} \pi (f(x) - b)^2 , dx ]
Similarly, revolving around a vertical line ( x = a ) looks like this: [ \int_{c}^{d} \pi (f(y) - a)^2 , dy ]
Washer Method 🧼
The Washer Method is like taking an Oreo and removing the cream. You’re left with two circular cookie pieces, one inside the other, creating a delectable washer shape. The integral is basically the difference in their “creamy” areas, summed over the interval.
The general equation for this work of art is: [ \int_{c}^{d} \pi \left( (f(x))^2 - (g(x))^2 \right) dx ]
Here, ( f(x) ) is the outer radius, and ( g(x) ) is the inner radius. Think of ( f(x) ) as your protective older sibling, and ( g(x) ) as you—the integral finds space between the two.
Washer Method Around Other Axes ↪️
When revolving around a line other than the x- or y-axis, we tweak our integral slightly. Using our newfound knowledge:
[ \int_{c}^{d} \pi \left( (f(x) - b)^2 - (g(x) - b)^2 \right) dx ]
Memorize this formula, stash it in your mental toolbox, and you’re all set!
Practice Makes Perfect ✏️
Example 1:
Find the volume of the solid by revolving the area between ( f(x) = -x^2 - 5 ) and ( g(x) = e^x ) around the line ( y = -2 ).
First, graph the functions. (Pro tip: Desmos helps visualize things beautifully!)
Now, let’s identify the upper and lower bounds by solving ( -x^2 - 5 = e^x ). Use a graphing calculator or software to find that these intersections approximately occur at ( x = -2.211 ) and ( x = 1.241 ).
Now to the integral: [ \int_{-2.211}^{1.241} \pi \left( (-x^2 - 5 + 2)^2 - (e^x + 2)^2 \right) dx ]
Simplify and solve: [ \int_{-2.211}^{1.241} \pi \left(( -x^2 - 3)^2 - (e^x + 2)^2 \right) dx ]
[ \int_{-2.211}^{1.241} \pi \left( x^4 - 6x^2 + 9 - (e^{2x} + 4e^x + 4) \right) dx ]
After calculating this integral using basic integration rules and a calculator, we conclude with a volume of approximately 275.7 cubic units.
Example 2:
Find the volume of the funnel with the radius at height ( h ) given by ( r = \frac{3 + h^2}{20} ), bounded between ( h = 0 ) and ( h = 10 ).
Given the radius function, set up the integral: [ \int_{0}^{10} \pi \left( \left(\frac{3 + h^2}{20}\right)^2 \right) dh ]
Simplify and solve: [ \frac{\pi}{400} \int_{0}^{10} (9 + 6h^2 + h^4) dh ]
Evaluate using basic integration: [ \frac{\pi}{400} \left[ 9h + 2h^3 + \frac{h^5}{5} \right]_{0}^{10} ]
This gives: [ \frac{\pi}{400} \left( 90 + 2000 + 2000 \right) = \frac{4090 \pi}{400} = \frac{409\pi}{40} , \text{cubic inches} ]
Key Terms to Know 📚
- Axis of Revolution: The line around which a shape rotates to form a 3D object. Think of it like the centerstage of a ballet performance.
- Bounded by Lines: Enclosed by straight boundaries.
- Definite Integral: Calculates exact area under a curve.
- Differential Element ( dA ): Tiny area that approximates the total surface area.
- Inner and Outer Radii: Distances from the axis to the curves.
- Integrate: Summing up infinitesimal areas under a curve.
- Washer Method: Volume by subtracting one function’s area from another.
- (\pi): The star of circles, approximately 3.14159…
Conclusion ⭐
So there you have it, folks! We've mastered the Washer Method and spiced things up by revolving around different axes. Remember to always visualize the problem, find your radii and bounds, and don’t forget to practice plenty. You've got the tools—now go show those calculus problems who's boss! 🎓📈
