Volume

Cross sections

Volume = area × height

integrable cross sections

Volume of a solid of integrable cross-sectional area $𝒜 (x)$ from $x = 𝒶$ to $x = 𝒷$ is

𝒱 = \int_{𝒶}^{𝒷} 𝒜 (x) d x

integrable cross sections

Sketch the solid and typical cross section region.
Formulate the area function $𝒜 (x)$ .
Find the limits of integration.
Evaluate the integral.

pyramid

Find the volume of a pyramid with height 12 and base 6.

wedge

A curved wedge is cut from a circular cylinder of radius three by two planes. One plane is perpendicular to the axis of the cylinder and the other forms a $\frac{π}{4}$ angle with the first one at the center of the cylinder. Find the volume of the wedge.

solid

Find the volume of a solid with base the region between the functions $f (x) = x^{2} - 1$ and $g (x) = - x^{2} + 1$ . Its cross sections perpendicular to the $x$ -axis are equilateral triangles.

Solids of Revolution

slices

\begin{matrix} 𝒱 & = & \int_{𝓁}^{𝓊} 𝒜 d ℎ \\ = & \int_{𝒶}^{𝒷} π {(ℛ (x))}^{2} d x \\ = & \int_{𝒸}^{𝒹} π {(ℛ (y))}^{2} d y \end{matrix}

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = x^{2} - 4 x + 5$ , $x = 1$ and $x = 4$ rotated around the $x$ -axis.

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = \sqrt{x}$ , from $x = 0$ to $x = 4$ rotated around the $x$ -axis.

Find the volume of the solid of revolution formed by revolving region bounded by $g (y) = \sqrt{4 - y}$ and the $y$ -axis over the $y$ -interval $[0, 4]$ .

Solids of Revolution

slices

Washer method

\begin{matrix} 𝒱 & = & \int_{𝓁}^{𝓊} 𝒜 d ℎ \\ = & \int_{𝒶}^{𝒷} π ({[ℛ (x)]}^{2} - {[𝓇 (x)]}^{2}) d x \\ = & \int_{𝒸}^{𝒹} π ({[ℛ (y)]}^{2} - {[𝓇 (y)]}^{2}) d y \end{matrix}

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = \sqrt{x}$ and $g (x) = 1$ rotated around the $x$ -axis over the $x$ -interval $[1, 4]$ .

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = x$ and $g (x) = \frac{1}{x}$ rotated around the $x$ -axis over the $x$ -interval $[1, 4]$ .

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = 4 - x$ and the $x$ -axis rotated around the line $y = -2$ over the $x$ -interval $[0, 4]$ .

Solids of Revolution

Cylindrical shells

\begin{matrix} 𝒱 & = & \int_{𝓁}^{𝓊} ℎ d 𝒜 \\ = & \int_{𝒶}^{𝒷} [f (x)] [2 π x d x] & = & \int_{𝒶}^{𝒷} 2 π x f (x) d x \\ = & \int_{𝒸}^{𝒹} [g (y)] [2 π y d y] & = & \int_{𝒸}^{𝒹} 2 π y g (y) d y \end{matrix}

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = x + 1$ and $g (x) = {(x - 1)}^{2}$ rotated around the $y$ -axis.

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = - x^{3} + 2 x^{2}$ in the first quadrant rotated around the $y$ -axis.

Find the volume of the solid of revolution formed by revolving region bounded by $f (x) = \sqrt{x}$ , from $x = 0$ to $x = 4$ rotated around the $x$ -axis.

Find the volume of the solid of revolution formed by revolving region bounded by $g (y) = \sqrt{4 - y}$ and the $y$ -axis over the $y$ -interval $[0, 4]$ .