Lecture #5: Centers of Mass & Volumes of Revolution

Archimedes’ Law of the Lever

Two magnitudes balance at distances reciprocally proportional to their magnitudes.

Centers of Gravity for Some Elementary 2D Shapes

Archimedes showed that:

• The center of gravity of a parallelogram is the point of intersection of the diagonals.
• The center of gravity of a triangle lies on the straight line between a vertex and the center of the opposite side.
• The center of gravity of a parabolic segment lies at a point \(G\) on the diameter of the parabola which is \(\frac{3}{5}\) of the way from the vertex to the base.

Center of Gravity

Given any region bounded by \(f(x)\) and \(g(x)\), can we find the coordinates \((\overline{x}, \overline{y})\) of its center of gravity?

The Moment Equation

If masses \(m_1, m_2, \dots, m_n\) are suspended on a balance beam from points \(x_1, x_2, \dots, x_n\), then the system will be in balance when suspended from the point \(\overline{x}\) such that:

\[ \sum_{i=1}^n m_i (\overline{x} - x_i) = 0, \]

where:

\[ \begin{aligned} \overline{x} = \dfrac{\displaystyle\sum_{i=1}^n m_i x_i}{\displaystyle\sum_{i=1}^n m_i}. \end{aligned} \]

Center of Gravity

\[ \begin{aligned} \overline{x} = \frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^n m_i} \quad\Longrightarrow \quad \overline{x}_P = \frac{\sum_{i=1}^n x_i^* (f(x_i^*) - g(x_i^*)) \Delta x_i}{\sum_{i=1}^n (f(x_i^*) - g(x_i^*)) \Delta x_i} \end{aligned} \]

Center of Gravity

If we send the width of the intervals to 0, we can rewrite these as intervals.

\[ \begin{aligned} \overline{x}_P &= \lim\limits_{\Delta x \to 0} \frac{\sum_{i=1}^n x_i^* (f(x_i^*) - g(x_i^*)) \Delta x_i}{\sum_{i=1}^n (f(x_i^*) - g(x_i^*)) \Delta x_i} \\[1em] &= \frac{\displaystyle\int_a^b x\left(f(x) - g(x)\right) \, dx}{\displaystyle\int_a^b f(x) - g(x) \, dx} \end{aligned} \]

Center of Gravity

\[ \begin{aligned} \overline{y} = \frac{\sum_{i=1}^n m_i y_i}{\sum_{i=1}^n m_i} \,\Longrightarrow \, \overline{y}_P = \frac{\sum_{i=1}^n \left( \frac{f(x_i^*) + g(x_i^*)}{2} \right) \left( f(x_i^*) - g(x_i^*) \right) \Delta x_i}{\sum_{i=1}^n (f(x_i^*) - g(x_i^*)) \Delta x_i} \end{aligned} \]

Center of Gravity

If we send the width of the intervals to 0, we can rewrite these as intervals.

\[ \begin{aligned} \overline{y}_P &= \lim\limits_{\Delta x \to 0} \frac{\sum_{i=1}^n \left( \frac{f(x_i^*) + g(x_i^*)}{2} \right) \left( f(x_i^*) - g(x_i^*) \right) \Delta x_i}{\sum_{i=1}^n (f(x_i^*) - g(x_i^*)) \Delta x_i} \\[1em] &= \frac{1}{2} \left( \frac{\displaystyle\int_a^b f(x)^2 - g(x)^2 \, dx}{\displaystyle\int_a^b f(x) - g(x) \, dx} \right) \end{aligned} \]

Center of Gravity

If a region \(R\) is bounded by the graphs of \(f(x) > g(x)\) on the interval \([a, b]\), then the center of gravity of \(R\) can be found at the point with coordinates \((\overline{x}, \overline{y})\), where:

\[ \begin{aligned} \overline{x} &= \frac{\displaystyle\int_a^b x\left(f(x) - g(x)\right) \, dx}{\displaystyle\int_a^b f(x) - g(x) \, dx} \\[0.5em] \overline{y} &= \frac{\displaystyle\int_a^b f(x)^2 - g(x)^2 \, dx}{2 \displaystyle\int_a^b f(x) - g(x) \, dx} \end{aligned} \]

Example: Center of Gravity

Example: Find the center of gravity of the region bounded by the curves \(y=1\) and \(y=x^2\).

Example: Center of Gravity

Example: Find the center of gravity of the region bounded by the curves \(y=\sqrt{x}\) and \(y=x^3\).

Volume of Revolution

Given a curve \(f(x) \ge 0\) over the interval \([a, b]\), a volume of revolution is the 3D figure obtained by revolving the region between \(f(x)\) and the \(x\)-axis about an axis.

Volume of Revolution

Example: \(f(x) = \sin(x)\) on the interval \(x \in [0, \pi]\).

How do we calculate the volume of these?

Volume of Revolution

Volume of Revolution

The Pappus-Guldin Theorem

Let \(R\) be a plane region that lies entirely on one side of a line \(l\) in the plane. If \(R\) is rotated about \(l\), then the volume of the resulting solid is the product of the area \(A\) of \(R\) and the distance traveled by the centroid of \(R\).

The Disk/Washer Method

Suppose \(f(x) \ge g(x) \ge 0\) over an interval \([a, b]\). Then the volume of revolution obtained by rotating the region contained by \(f(x)\) and \(g(x)\) over the interval \([a,b]\) around the \(x\)–axis is:

\[ V = \pi \int_a^b f(x)^2 - g(x)^2 \, dx \]

\[ \begin{aligned} V &= (2 \pi \overline{y}) \times A_R \\ &= 2\pi \left( \frac{\displaystyle\int_a^b f(x)^2 - g(x)^2 \, dx}{2 \displaystyle\int_a^b f(x) - g(x) \, dx} \right) \left( \int_a^b f(x) - g(x) \, dx \right) \end{aligned} \]

Example: The Disk/Washer Method

Example: Find the volume of revolution obtained by revolving the region beneath the curve \(y = x^2\) around the \(x\)–axis with \(0 \le x \le 1\).

The Shell Method

Suppose \(f(x) \ge g(x) \ge 0\) over an interval \([a, b]\). Then the volume of revolution obtained by rotating the region contained by \(f(x)\) and \(g(x)\) over the interval \([a,b]\) around the \(y\)–axis is:

\[ V = 2\pi \int_a^b x \left( f(x) - g(x) \right) \, dx \]

\[ \begin{aligned} V &= (2 \pi \overline{x}) \times A_R \\ &= 2\pi \left( \frac{\displaystyle\int_a^b x\left(f(x) - g(x)\right) \, dx}{\displaystyle\int_a^b f(x) - g(x) \, dx} \right) \left( \int_a^b f(x) - g(x) \, dx \right) \end{aligned} \]

Example: The Shell Method

Example: Find the volume of revolution obtained by revolving the region beneath the curve \(y = x^2\) around the \(x\)–axis with \(0 \le y \le 1\).