Question

Find the total mass of the triangular region shown below. All lengths are in cm, and the density of the region is sigma(x) = 1 + x grams/cm^2

Answer 1

Answer:

3 grams

Step-by-step explanation:

We are going to take the mass of a bunch of little strips below the triangle "roof." To do this, we must figure out what formula for the mass we'll use, in this case, we'll use:

Mass of strip = denisty * area = (1+x)*y*deltax grams

now, because the "roof" of the triangle contains two different integrals (it completely changes direction), we will use TWO integrals!

**pretend ∈ is the sum symbol

Mass of left part = lim x->0 ∈ (1+x)*y*deltax = inegral -1 to 0 of (1+x)*3*(x+1) = 3 * integral -1 to 0 of (x^2 + 2x + 1) = 3 *  1/3 = 1

Mass of left part = lim x->0 ∈ (1+x)*y*deltax = inegral 0 to 1 of (1+x)*3*(-x+1) = 3 * integral 0 to 1 of (-x^2 + 1) = 3 * 2/3 = 2

Total mass = mass left + mass right = 1 + 2 = 3 grams