Find the mass of the triangular area with vertexes the points (1,-1), (4,1) and (4,3) if the density is equal to d(x,y)=x^2.
Can you help me solve this problem??? It's an exams theme
Hey rikelda91.
The first thing you need to do is setup the integral. You have a triangle with given vertices so you will need a double integral.
The definition of mass (in this case) is Integral (over A) d(x,y) dA where A is the region of the triangle. (Usually density is with respect to volume).
Hint: Since d(x,y) is a function of x, I'd suggest you formulate the region A in terms of two triangles which have an edge parallel to the y-axis. (It will help you a lot if you draw a diagram on paper and separate the triangles in the way I've mentioned).
I don't see any good reason to divide the triangle into two parts like that. x goes from 1 to 4 and, for each x, y goes from the lower line to the upper. those are, as you say, y= (4/3)x- 7/3 and y= (2/3)x- 5/3. Integrate $\displaystyle f(x,y)= x^2$: $\displaystyle \int_{x= 1}^4\int_{y= (2/3)x-5/3}^{(4/3)x- 7/3} x^2 dydx$.