§6.2 Using definite integrals to find volume

Introduction

The Volume of a Solid of Revolution

Revolving about the

Consider a circular cone of radius 3 and height 5, which we view horizontally as pictured in Figure. Our goal in this activity is to use a definite integral to determine the volume of the cone.

Find a formula for the linear function $y = f(x)$ that forms the top “edge” of the cone that is pictured in Figure. That is, find a formula for the linear function that connects the points $(0,3)$ and $(5,0)$ to form the segment that is used to create the cone as a solid of revolution.

Find the volume of the solid of revolution generated when the region $R$ bounded by $y = 4-x^2$ and the $x$-axis is revolved about the $x$-axis.

Find the volume of the solid of revolution generated when the finite region $R$ that lies between $y = 4-x^2$ and $y = x+2$ is revolved about the $x$-axis.

The region $S$ bounded by the $x$-axis, the curve $y = \sqrt{x}$, and the line $x = 4$; revolve $S$ about the $x$-axis.

The region $S$ bounded by the $y$-axis, the curve $y = \sqrt{x}$, and the line $y = 2$; revolve $S$ about the $x$-axis.

The finite region $S$ bounded by the curves $y = \sqrt{x}$ and $y = x^3$; revolve $S$ about the $x$-axis.

The finite region $S$ bounded by the curves $y = 2x^2 + 1$ and $y = x^2 + 4$; revolve $S$ about the $x$-axis.

The region $S$ bounded by the $y$-axis, the curve $y = \sqrt{x}$, and the line $y = 2$; revolve $S$ about the $y$-axis. How is this problem different from the one posed in part (b)?

Find the volume of the solid of revolution generated when the finite region $R$ that lies between $y = \sqrt{x}$ and $y = x^4$ is revolved about the $y$-axis.

The region $S$ bounded by the $y$-axis, the curve $y = \sqrt{x}$, and the line $y = 2$; revolve $S$ about the $y$-axis.

The region $S$ bounded by the $x$-axis, the curve $y = \sqrt{x}$, and the line $x = 4$; revolve $S$ about the $y$-axis.

The finite region $S$ in the first quadrant bounded by the curves $y = 2x$ and $y = x^3$; revolve $S$ about the $x$-axis.

The finite region $S$ in the first quadrant bounded by the curves $y = 2x$ and $y = x^3$; revolve $S$ about the $y$-axis.

The finite region $S$ bounded by the curves $x = (y-1)^2$ and $y = x-1$; revolve $S$ about the $y$-axis