AP Calculus AB: Lesson 7.6: Volume using Known Cross Sections

Michele Chromel introduces the concept of finding the volume of a solid with known cross-sectional shapes. The lesson begins with a scenario involving a 3D solid and parabolic cross-sections perpendicular to the x-axis, each with a thickness represented by Δx. The volume of a single slice is calculated as the area of the cross-section (a(x_k)) times the thickness. The process involves summing these volumes over multiple slices using a definite integral from x=a to x=b, where the area function a(x) is given. An example is provided with a region R enclosed by y=√x, y=x^2, and the x-axis, with cross-sections having an area given by sin(π/2 * x). The integral set-up for the volume calculation is demonstrated.

The script explains the use of u-substitution in integrals to calculate the volume of a solid. The example involves integrating the sine function with respect to u, where u = π/2 * x, and the bounds of integration are adjusted accordingly. The integral of -cos(u) from 0 to π is computed, resulting in a volume calculation that simplifies to 4/π. Interactive demonstrations are recommended for visualizing 3D solids, with three different websites provided for exploring cross-sections of various shapes.

The video script describes how to visualize 3D solids with different cross-sectional shapes such as squares, equilateral triangles, and semicircles. It explains the process of adjusting sliders to change the number of cross-sections (N value) and the interval end (x value) to better understand the solid's volume. The base of the solid is represented by a region between two functions, and the cross-sections' side lengths are determined by the vertical distance between these functions. The script also demonstrates how to change the shape of the cross-sections from squares to triangles to semicircles using checkboxes.

The script details the process of setting up integrals for calculating the volume of solids with various cross-sectional shapes. It explains how to find the area of a cross-section, such as a square or an equilateral triangle, and how to multiply this area by the thickness (dx) to find the volume of a single slice. The integral of this volume expression over the interval of interest is then used to find the total volume of the solid. The process is illustrated with examples of solids with bases defined by different functions and cross-sections of squares and equilateral triangles.

The script continues with the calculation of solid volumes where the cross-sections are isosceles right triangles and semicircles. For isosceles right triangles, the area function is derived from the properties of a 45-45-90 triangle, and the volume is found by integrating the area function over the interval. For semicircles, the area function is one-fourth of the area of a full circle, and the volume is calculated similarly. The process involves identifying the correct area function for the cross-section and integrating it over the specified interval.

The script addresses more complex scenarios where two functions define the base of the region, and the cross-sections are squares or equilateral triangles. It demonstrates finding the points of intersection between the functions and setting up the integral for the volume of the solid. The process includes squaring the difference of the functions to find the area of the cross-sections and integrating this area over the interval to find the volume. The calculations are shown for both square and equilateral triangle cross-sections.

The final part of the script presents an example where the cross-sections are quarter circles, defined by two functions. The area function for a quarter circle is derived, and the volume of the solid is calculated by integrating this area function over the interval from x=-1 to x=2. The script concludes with the integral calculation and the final volume expressed in terms of π, resulting in a value of 81π/40 after simplification.

The script concludes with a prompt for critical thinking about the process of finding the volume of solids with known cross-sections, emphasizing the importance of understanding the process over memorizing formulas. It encourages reflection on the steps taken throughout the lesson for setting up integrals and calculating volumes.