91Ó°ÊÓ

The mass of a rectangular block of metal that is 30 centimeters wide, 21 centimeters long, and 14 centimeters tall . its density y centimeters from its base is

$\mathrm{ÒÏ}(y)=6.8+0.14y$

Volume of cuboid = length times width times height

The length and height of cuboid is 21 cm and height is 14 cm. The width is $∆y$

$$\begin{gathered} Volume=\left(21\right)\left(14\right)∆y \\ V=294∆y \end{gathered}$$

The density of entire slice is role="math" localid="1649436313176" ${\mathrm{ÒÏ}}_k(y)=6.8+0.14y_k$

Density formula is

role="math" localid="1649436403801" $$\begin{gathered} \mathrm{ÒÏ}=\frac{M}{V} \\ 6.8+0.14y_k=\frac{M}{294∆y} \\ M=\left(6.8+0.14y_k\right)294∆y \end{gathered}$$

Approximate mass of the entire rod is the sum

$\underset{}{∑}\left(6.8+0.14y_k\right)294∆y$

Accumulate the mass from y=0 to y=10 to find total mass of the rod

role="math" localid="1649436751462" $$\begin{gathered} {∫}_0^{10}\left(6.8+0.14y\right)294∆y \\ =294{∫}_0^{30}\left(6.8+0.14y\right)∆y \\ =294{(6.8y+0.07y^2)}_0^{30} \\ =294(6.8(30)+0.07{\left(30\right)}^2) \\ =78498grams \end{gathered}$$