Explore a comprehensive series of calculus lectures focusing on volume calculations and related applications. Dive into various methods including slicing, cylindrical shells, and similar cross-sections to determine volumes of complex shapes. Learn how to apply these techniques to rotated curves, spheres, pyramids, and more. Discover Newton's Method and its potential pitfalls. Gain insight into choosing the most efficient approach for different volume problems. Practice with exam-style questions and worked examples to reinforce your understanding. Investigate Simpson's Rule for volume approximation and explore the connections between volume calculations and other areas of mathematics.
Overview
Syllabus
Introduction to Volumes by Slicing.
Volumes by Slicing: Understanding the Annulus.
Volumes by Slicing: Volume Generated by Rotation About y = 6.
Volumes by Slicing: Rotation around x = 1.
Calculating a Volume Rotated Around x = y (1 of 2: Determining Radius).
Calculating a Volume Rotated Around x = y (2 of 2: Forming/Evaluating Integral).
Applications & Implications of d/dx(½v²): General Case.
Evaluating a Volume by Slices & by Shells.
Introduction to Volumes by Cylindrical Shells: Visual Comparison with Slicing.
Generalising from Volumes by Slices & Shells.
Introduction to Volumes by Similar Cross-Section: Square Pyramid.
Volumes by Cross-Section: Circular Slanted Roof.
Useful Tricks for Evaluating Integrals from a Volume.
Volume of a Sphere: Three Different Derivations.
Volume of a Tetrahedron (by similar cross-sections).
Volumes by Cylindrical Shells (example question from exam).
Volumes by Slices (example question from exam: hole drilled through sphere).
Newton's Method (1 of 2: How does it work?).
Newton's Method (2 of 2: Potential Dangers).
Volumes (Ext II) (How does Volumes Fit in with the other mathematics courses).
Volumes by Slicing (1 of 4: Proving the Volume formula integral).
Volumes by Slicing (2 of 4: Finding the Volume of a Typical Slice to calculate a volume).
Volumes by Slicing (3 of 4: Rotating an area around an axis apart from coordinate axis).
Volumes by Slicing (4 of 4: Harder Volumes by Slicing Question).
Harder Volumes by Slicing (1 of 3: Finding the orientation of slices and volume of typical slice).
Harder Volumes by Slicing (2 of 3: Using a triangle to find x in terms of a defined variable 'h').
Harder Volumes by Slicing (3 of 3: Converting to one single variable to integrate for the volume).
Volume by Cylindrical Shells (3 of 3: Finding the Volume via Cylindrical Shells and which to choose).
Volumes by Shells (1 of 3: Overview of Volumes done so far).
Volumes by Shells (2 of 3: Introduction to Cylindrical Shells & finding typical volume).
Worked Example of Volumes by shells (Finding the volume of an area between two curves).
Choosing between Slices and Shells (1 of 2: Volume when cosx is rotated around y axis by slicing).
Choosing between Slices and Shells (2 of 2: Finding the volume and the benefits of shell method).
Volumes by Similar Cross Sections (1 of 2: Using integration to find a non-rotated volume).
Volumes by Similar Cross Sections (2 of 2: Using a typical cross section volume to find the volume).
Volumes by Similar Cross Section (1 of 4: Presenting information from the question with a diagram).
Volumes by Similar Cross Section (2 of 4: Finding the area of a typical volume).
Volumes by Cross Section (4 of 4: Finding the side length in term of h and finding the volume).
Volumes by Similar Cross Section (3 of 4: Finding the volume of a double quarter pipe).
Simpson's Rule for Volumes.
Taught by
Eddie Woo
