The volume of a solid with known cross sections can be calculated by taking the definite integral of all the cross sections, with being equal to a single section.
- CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes Number of questions—2. A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height h feet is given. The function A, where. Ah is measured in square feet.
- So whatever this length is, we also go that much high, and so the cross section is a square right over there. The cross section right over here is going to be a square. Whatever the difference between these two functions is, that's also how high we are going to go. This length, which is six at this point, this is also going to be the height.
- Calculus Chapters 7 & 8. Chapter 7 Resources. Chapter 7 unassigned problems that would be good review for test: 181, 183-186, 194, 195, 198. Video Cross-Sectional.
For example, suppose we want to find the volume of the solid with each cross section being a circle, with the diameter of each cross section being the distance between and the x-axis from 0 to 2. Since one cross section will be equal to and , a single cross section will have an area of or . The volume will therefore be equal to
This method can be used to derive geometric formulas. Here, we will find the volume of a pyramid with sides of the base equal to and height . Since , all that is needed is to find a function of to describe the area of a cross section of the pyramid at height . Using trigonometry, we can find that . All that is needed is to isolate so that and square it (since this function describes the area of a cross section), then integrate from to . It is important to remember that and are constants and must be treated as such.
Table of Contents and Introduction: General Overview of Calculus Chapter 1: Review of Derivatives Chapter 1 pdf 1.1: The Power and Exponential Rules with the Chain Rule; 1.2: Trig, Trig Inverse, and Log Rules; 1.3: Local Linearity, Euler’s Method, and Approximations. The AP Calculus Problem Book Publication history: First edition, 2002 Second edition, 2003 Third edition, 2004 Third edition Revised and Corrected, 2005.
7.8 Volumes With Cross Sectionsap Calculus 14th Edition