Calculus IV: Vector Calculus (Line Integrals, Surface Integrals, Vector Fields, Greens' Thm, Divergence Thm, Stokes Thm, etc) **Full Course** Why is the surface area of a Sphere 4pi times radius squared???
Why is the surface area of a Sphere 4pi times radius squared??? Transcript and Lesson Notes
You might have memorized that the surface area of a sphere of radius a is 4 pi a^2. But why? In this video we will compute it using the surface area formula we came up with in the previous video. First, we will parametri
Quick Summary
You might have memorized that the surface area of a sphere of radius a is 4 pi a^2. But why? In this video we will compute it using the surface area formula we came up with in the previous video. First, we will parametri
Key Takeaways
- Review the core idea: You might have memorized that the surface area of a sphere of radius a is 4 pi a^2. But why? In this video we will compute it using the surface area formula we came up with in the previous video. First, we will parametri
- Understand how Solution fits into Why is the surface area of a Sphere 4pi times radius squared???.
- Understand how Example fits into Why is the surface area of a Sphere 4pi times radius squared???.
- Understand how math fits into Why is the surface area of a Sphere 4pi times radius squared???.
- Understand how Surface Area fits into Why is the surface area of a Sphere 4pi times radius squared???.
Key Concepts
Full Transcript
You might have memorized that the surface area of a sphere of radius a is 4 pi a^2. But why? In this video we will compute it using the surface area formula we came up with in the previous video. First, we will parametrize the surface using spherical coordinates with a fixed radius. Second, we will plug it into the surface area formula for parameterized surfaces that we derived in the previous video in the playlist (see below). At this point it is just computational to compute the length of the cross product and evaluate the resulting double integral. Note that this video is quite computational, it doesn't shed any light into the geometry of why this makes sense, it just says we get this result when we plug into the formulas we derived previously. For a more geometric view interpretting this as 4 times the shadow of pi*radius^2 check out the 3blue1brown video here: https://www.youtube.com/watch?v=GNcFjFmqEc8&ab_channel=3Blue1Brown MY VECTOR CALCULUS PLAYLIST: ►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa OTHER COURSE PLAYLISTS: ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►DIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Cool Math Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett SOCIALS: ►Twitter (math based): http://twitter.com/treforbazett ►Instagram (photography based): http://instagram.com/treforphotography
Lesson FAQs
What is Why is the surface area of a Sphere 4pi times radius squared??? about?
You might have memorized that the surface area of a sphere of radius a is 4 pi a^2. But why? In this video we will compute it using the surface area formula we came up with in the previous video. First, we will parametri
What key concepts are covered in this lesson?
The lesson covers Solution, Example, math, Surface Area, Sphere.
What should I learn before Why is the surface area of a Sphere 4pi times radius squared????
Review the previous lessons in Calculus IV: Vector Calculus (Line Integrals, Surface Integrals, Vector Fields, Greens' Thm, Divergence Thm, Stokes Thm, etc) **Full Course**, then use the transcript and key concepts on this page to fill any gaps.
How can I practice after this lesson?
Practice by applying the main concepts: Solution, Example, math, Surface Area.
Does this lesson include a transcript?
Yes. The full transcript is visible on this page in indexable HTML sections.
Is this lesson free?
Yes. CourseHive lessons and courses are available to learn online for free.
Continue Learning
Continue this lesson in the app
Install CourseHive on Android or iOS to keep learning while you move.
