Volume Using Known Cross Sections (Slicing) | Calculus 2 Lesson 5 - JK Math

The video begins with an introduction to finding the volume of solids with known cross sections using definite integrals. It contrasts the disk method, which is used for solids of revolution with circular cross sections, with a more general approach that can handle other shapes. The importance of choosing the correct axis (x or y) for integration based on the orientation of the cross sections is emphasized. Examples are provided to illustrate how to calculate the volume of solids with square and semi-circular cross sections.

The video then lists common cross-section shapes—squares, semicircles, rectangles, equilateral triangles, and isosceles triangles—along with their respective area formulas. It highlights the need to memorize these formulas for different cross-section shapes, especially the area formula for an equilateral triangle, which is less commonly used. The video prepares viewers for example problems by ensuring they understand how to calculate the area of various cross sections.

The first example demonstrates calculating the volume of a solid with a base bounded by the functions y = √x, x = 9, and y = 0, with cross sections that are squares perpendicular to the x-axis. The video shows how to graph the region representing the base of the solid and how to set up the definite integral to calculate the volume. The cross-sectional area is determined by the distance from the x-axis to the function y = √x, and the volume is found by integrating this area from x = 0 to x = 9.

The second example focuses on a solid with a triangular base bounded by y = (1/2)x, y = 2, and x = 0, with semi-circular cross sections perpendicular to the y-axis. The video illustrates the graphing of the base region and the setup for the definite integral using the area of a semicircle. The radius of the semicircles is determined by half the distance from the y-axis to the boundary line y = (1/2)x. The volume is calculated by integrating the area of the semicircles from y = 0 to y = 2.

The third example involves a solid with a base bounded by y = 4 - x, y = x - 1, and x = 0, with equilateral triangular cross sections perpendicular to the x-axis. The video explains how to graph the base region and determine the length of the sides of the equilateral triangles, which is the distance between the two functions y = 4 - x and y = x - 1. The cross-sectional area is calculated using the area formula for an equilateral triangle, and the volume is found by integrating this area from the lower bound x = 0 to the upper bound where the two functions intersect, x = 5/2.

The final part of the video completes the calculation for the volume of the solid with equilateral triangular cross sections. The integral is simplified and solved, resulting in the volume of the solid. The video concludes with an invitation for viewers to check out additional examples and to leave any questions in the comments section. The host signs off, indicating the end of the lesson.