8.8 Volumes with Cross Sections: Triangles and Semicircles

Volumes with cross sections can be a tricky concept to understand, but it is an important one for understanding calculus. In this guide, we will focus on how to find the volume of a solid with triangular and semicircular cross sections. First, let's define what is meant by a solid with cross sections. A solid with cross sections is a three-dimensional object that can be divided into many smaller two-dimensional shapes, such as triangles or semicircles. The volume of the solid is found by adding up the volumes of all of these smaller shapes. To find the volume of a solid with triangular or semicircular cross sections, we will use the method of integration. This method involves breaking the solid into many small slices, each with a triangular or semicircular cross section, and then finding the volume of each slice. The total volume of the solid is found by adding up the volumes of all of these slices. Here are the steps to find the volume of a solid with triangular or semicircular cross sections: Determine the function that describes the cross section of the solid. This function will be in the form of y = f(x) for triangular cross sections and y = √(r^2 - x^2) for semicircular cross sections, where r is the radius of the semicircle. Determine the limits of integration. These limits will be the range of x or y values for which the cross section is defined. Find the area of one cross section by plugging in the limits of integration into the function for the cross section. Use the definite integral to find the total volume of the solid by integrating the function for the cross section with respect to x or y and then multiplying by the width or height of the slice.

Example 4: Consider a solid with triangular cross sections, where the function describing the cross section is y = 3x. The limits of integration are from x = 0 to x = 2. To find the area of one cross section, we plug in x = 0 and x = 2 into the function and find that the area is 6. To find the total volume of the solid, we integrate the function with respect to x and multiply by the width of the slice, which is dx. This gives us the definite integral from 0 to 2 of 3x dx, which equals 6.

Example 5: Consider a solid with semicircular cross sections, where the radius of the semicircle is 1. The limits of integration are from x = 0 to x = 1. To find the area of one cross section, we plug in x = 0 and x = 1 into the function y = √(1^2 - x^2) and find that the area is (π * 1^2)/2 = π/2. To find the total volume of the solid, we integrate the function with respect to x and multiply by the width of the slice, which is dx. This gives us the definite integral from 0 to 1 of √(1^2 - x^2) dx, which equals (π * 1^3)/3 = (π)/3.

Example 6: Consider a solid with triangular cross sections, where the function describing the cross section is y = x^3. The limits of integration are from x = 0 to x = 1. To find the area of one cross section, we plug in x = 0 and x = 1 into the function and find that the area is 1/4. To find the total volume of the solid, we integrate the function with respect to x and multiply by the width of the slice, which is dx. This gives us the definite integral from 0 to 1 of x^3 dx, which equals 1/4.

Note that the examples provided are just a small sample of the different types of solids with cross sections that you can encounter, and the method for finding the volume remains the same regardless of the specific function or limits of integration. It is important to be comfortable with integration and to have a strong understanding of the fundamental concepts of calculus in order to be able to solve these types of problems.

It is important to note that when finding the volume of a solid with cross sections, the cross sections must be perpendicular to the axis of revolution. This means that the cross sections must be parallel to the x or y axis depending on the axis of revolution. In addition, the cross sections must also be of constant width or height.

By following these steps and examples, you should now have a good understanding of how to find the volume of a solid with triangular or semicircular cross sections using the method of integration. Remember to always pay attention to the function describing the cross section, the limits of integration, and the axis of revolution when finding the volume of a solid with cross sections.