- A box lies inside the rectangle $ [2,8]\times [1,3] $ (so $2\leq x\leq 8$ and $1\leq y \leq 3$ ). Compute the integral formula $$\frac{\int_2^8\int_1^3 x dydx}{\int_2^8\int_1^3 1 dydx}.$$ What physical quantity does the integral above compute?
- Draw the region described the bounds of each integral.
- $\ds\int_{0}^{2}\int_{2x}^{4}dydx$ and $\ds\int_{0}^{4}\int_{0}^{y/2}dxdy$
- $\ds\int_{-3}^{3}\int_{0}^{9-x^2}\int_{0}^{5}dzdydx$
- $\ds\int_{0}^{1}\int_{0}^{1-z}\int_{0}^{\sqrt{1-x^2}}dydxdz$
- Set up an integral formula to compute each of the following:
- The mass of a disc that lies inside the circle $x^2+y^2=9$ and has density function given by $\delta = x+10$
- The $x$-coordinate of the center of mass (so $\bar x$) of the disc above.
- The $z$-coordinate of the center-of-mass (so $\bar z$) of the solid object in the first octant (all variables positive) that lies under the plane $2x+3y+6z=6$.
- The $y$-coordinate of the center-of-mass (so $\bar y$) of the same object.
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