This chapter introduces four techniques for calculating the volume of various geometric objects. The first technique, the slicing method, involves cutting the object into thin slices and summing their volumes using a single integral. The second technique, associated with solids of revolution, uses two integrals: one to compute the area of a slice and the other to sum these areas over the object's extent. The third technique employs three integrals to sum the volume of an object. The fourth technique is not detailed in this excerpt. The chapter specifically focuses on the slicing technique for solids of revolution. It begins by recalling the formula for the surface area of a swept surface, which is given by \( S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + (\frac{dy}{dx})^2} \, dx \). The volume of the solid of revolution is then derived as \( V = \pi \int_{a}^{b} (f(x))^2 \, dx \). The text explains that when a solid of revolution is divided into thin disks, each disk has a volume of \( V_i = \pi ( f(x_i) )^2 \Delta x \). As the number of disks approaches infinity, the total volume is given by the integral \( V = \pi \int_{a}^{b} ( f(x) )^2 \, dx \). An example is provided to compute the volume of a cylinder, where the radius \( r \) and height \( h \) are used. The volume of the cylinder is calculated as \( V = \pi r^2 h \).This chapter introduces four techniques for calculating the volume of various geometric objects. The first technique, the slicing method, involves cutting the object into thin slices and summing their volumes using a single integral. The second technique, associated with solids of revolution, uses two integrals: one to compute the area of a slice and the other to sum these areas over the object's extent. The third technique employs three integrals to sum the volume of an object. The fourth technique is not detailed in this excerpt. The chapter specifically focuses on the slicing technique for solids of revolution. It begins by recalling the formula for the surface area of a swept surface, which is given by \( S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + (\frac{dy}{dx})^2} \, dx \). The volume of the solid of revolution is then derived as \( V = \pi \int_{a}^{b} (f(x))^2 \, dx \). The text explains that when a solid of revolution is divided into thin disks, each disk has a volume of \( V_i = \pi ( f(x_i) )^2 \Delta x \). As the number of disks approaches infinity, the total volume is given by the integral \( V = \pi \int_{a}^{b} ( f(x) )^2 \, dx \). An example is provided to compute the volume of a cylinder, where the radius \( r \) and height \( h \) are used. The volume of the cylinder is calculated as \( V = \pi r^2 h \).