Question

Find the total mass of the triangular region shown below. All lengths are in centimeters, and the density of the region is d(x)=5+x grams/cm^2.

Answer 1

(1)

Divide the region into vertical rectangular strips.

(2)

Find the area of a typical strip:

$$ \begin{array}{l} A=(y) \mathrm{d} x=(4 x+4) \mathrm{d} x=4(x+1) \mathrm{d} x, \quad-1 \leq x \leq 0 \\ A=(y) \mathrm{d} x=(-4 x+4) \mathrm{d} x=4(1-x) \mathrm{d} x, \quad 0 \leq x \leq 1 \end{array} $$

(3)

Find the mass of a typical strip:

$$ \begin{array}{l} -1 \leq x \leq 0 \\ m=A d(x)=4(5+x)(x+1) \mathrm{d} x=4\left(x^{2}+6 x+5\right) \mathrm{d} x \\ 0 \leq x \leq 1 \\ m=A d(x)=4(5+x)(1-x) \mathrm{d} x=4\left(5-4 x-x^{2}\right) \mathrm{d} x \end{array} $$

(4)

Find the total mass of the region:

$$ M=\int_{-1}^{0} 4\left(x^{2}+6 x+5\right) \mathrm{d} x+\int_{0}^{1} 4\left(5-4 x-x^{2}\right) \mathrm{d} x $$

$$ \begin{array}{l} =\frac{28}{3}+\frac{32}{3} =\frac{60}{3} \\ =20 \end{array} $$