
# 7.6E: Exercises


## Exercise $$\PageIndex{1}$$: Mass

In the following exercises, the region $$R$$ occupied by a lamina is shown in a graph. Find the mass of $$R$$ with the density function $$\rho$$.

1. $$R$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(6,0); \space \rho (x,y) = xy$$.

$$\frac{27}{2}$$

3. $$R$$ is the triangular region with vertices $$(0,0), \space (1,1)$$, and $$(0,5); \space \rho (x,y) = x + y$$.

4. $$R$$ is the rectangular region with vertices $$(0,0), \space (0,3), \space (6,3)$$ and $$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

$$24\sqrt{2}$$

5. $$R$$ is the rectangular region with vertices $$(0,1), \space (0,3), \space (3,3)$$ and $$(3,1); \space \rho (x,y) = x^2y$$.

6. $$R$$ is the trapezoidal region determined by the lines $$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, and $$x = 0; \space \rho (x,y) = 3xy$$.

$$76$$
7. $$R$$ is the trapezoidal region determined by the lines $$y = 0, \space y = 1, \space y = x$$ and $$y = -x + 3; \space \rho (x,y) = 2x + y$$.
8. $$R$$ is the disk of radius $$2$$ centered at $$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.