§6.3 Density, mass, and center of mass

Introduction

Density

Weighted Averages

Traffic density $c(x) = 200+100e^{-0.1x}$ is measured in cars per mile at $x$ miles east of an interchange. The quantity $c(x)\Delta x$ and the integral $\int_0^2c(x)\,dx$ are measured in

Evaluate $\int_0^2(200+100e^{-0.1x})\,dx$ to find how many cars occupy the two mile stretch east of the interchange. The exact value is $1400-1000e^{-0.2}$. Report the decimal, rounded to two places.

A 10 cm rod with cross section 1 cm$^2$ has density $\rho(x) = 2e^{-0.2x}$ g/cm. Its exact mass is $10-10e^{-2}$ grams. Report the decimal, rounded to three places.

A cone of base radius 4 m and height 5 m, viewed with its axis along the $x$-axis, is generated by the line $y = 4-\frac{4}{5}x$. Its volume equals

The same cone with volume $\frac{80\pi}{3}$ m$^3$ has uniform density 800 kg/m$^3$. Compute its mass in kilograms, rounded to three decimals. The exact value is $\frac{64000\pi}{3}$.

A 12 cm rod has density $\rho(x) = \frac{1}{25}(x-15)^2$ g/cm. To split it into two pieces of equal mass, we solve $\int_0^z\rho = \frac{1}{2}\int_0^{12}\rho$. This gives

Two books of weight 1 lb sit on a 6-foot shelf at $x_1 = 0$ and $x_2 = 6$. The balancing point $\overline{x}$ equals

Four 1-lb books sit on a 6-foot shelf at $x_1 = 0$, $x_2 = 2$, $x_3 = 4$, and $x_4 = 6$. The balancing point $\overline{x}$ is

Keep four 1-lb books but move the third to $x_3 = 3$, so the locations are $0, 2, 3, 6$. Compute the new balancing point as a decimal.

Books of weight 2 lb at $x_1 = 0$, 3 lb at $x_2 = 2$, 1 lb at $x_3 = 4$, and 1 lb at $x_4 = 6$ sit on a shelf. Compute the balancing point $\overline{x}$ as a decimal, rounded to three places.

Keeping the weights 2, 3, 1, 1 but moving the third book to $x_3 = 5$, compute the new balancing point as a decimal, rounded to three places.

Keep the locations $0, 2, 4, 6$ but raise the third book's weight to 2 lb, so the weights are 2, 3, 2, 1. Compute the balancing point as a decimal.

Starting from weights 2, 3, 1, 1 at locations $0, 2, 4, 6$ with balance $\frac{16}{7}$, which single change moves the balancing point to the left?

Moving one book to the left moves the balancing point to the left, and adding weight at a location right of the current balancing point moves the balancing point to the right.