Hello, Tutors!
I am struggling on how to do problems related to finding the volume using integrals. Could you please help me?
Any easy-to understand resources?
Thank You for all your help. I am thankful for you tutors!!!!!!!
google is your friend.
lots of examples online.
you looking for solids of revolution, or 3D volumes in general?
I am looking for 3 D volumes. Finding A(y) through similar triangles is difficult in terms of 3 D. I still do not understand how to do them despite looking at the book examples.
I know that after finding A(y) I have to integrate it to find the volume but the first step is to find A(y), the hardest part for me.
Hello! I'm here to help you with finding the volume using integrals. Understanding this concept can be tricky at first, but with practice and some helpful resources, you'll be able to tackle those problems with confidence.
To find the volume of a solid using integrals, you need to first understand what type of solid you're dealing with and how it is related to the integral. There are three common methods to find volume: the disk method, the shell method, and the method of cross sections.
1. Disk Method:
This method is typically used when you're rotating a region around the x-axis or y-axis to create a solid. The volume is found by integrating the areas of infinitely many disks or washers that make up the solid. To get started with the disk method, here are some steps to follow:
- Identify the region that will be rotated.
- Determine whether the solid is being formed by rotating around the x-axis or y-axis.
- Determine the limits of integration (x or y) based on the given problem.
- Find the radius of each disk or washer using the distance from the axis of rotation.
- Integrate the area function with respect to x or y depending on the axis of rotation.
- Evaluate the definite integral to obtain the volume.
2. Shell Method:
The shell method is used when you're rotating a region around a vertical or horizontal line to create a solid. This method involves integrating the volume of infinitesimally thin cylindrical shells. Here are some steps to follow:
- Identify the region that will be rotated.
- Determine the axis of rotation and the direction of integration (vertical or horizontal).
- Determine the limits of integration based on the given problem.
- Find the radius of each shell and the height of each shell.
- Integrate the volume function with respect to x or y depending on the axis of rotation.
- Evaluate the definite integral to obtain the volume.
3. Method of Cross Sections:
The method of cross sections is used when you're given a solid with a known cross-sectional area. This method involves integrating the areas of the cross sections to find the volume. Here are some steps to follow:
- Identify the region for which you're given the cross-sectional area.
- Determine the axis of integration (x or y) based on the given problem.
- Find the area function for each cross section as a function of x or y.
- Integrate the area function with respect to x or y.
- Evaluate the definite integral to obtain the volume.
To further enhance your understanding and gain practice, I recommend the following resources:
- Online tutorials: Websites like Khan Academy, MIT OpenCourseWare, and Paul's Online Math Notes offer free math tutorials and practice problems. They usually have dedicated sections for calculus, including volumes of solids.
- Textbooks: Refer to your calculus textbook for detailed explanations, examples, and practice problems related to finding volume using integrals.
- YouTube videos: Search for videos that explain volume using integrals. There are many channels like PatrickJMT, The Organic Chemistry Tutor, and Michel van Biezen, which provide step-by-step explanations.
- Online forums and communities: Join online math forums or communities where you can interact with fellow students, ask questions, and get help from others who have mastered the topic.
Remember, practice is key to mastering any concept in calculus. Work through a variety of problems and seek clarification whenever you get stuck. Don't hesitate to ask for help from your tutor or fellow students. You can do it!