Triple integrals

Exercise

1.

For $${T = \{(x,y,z) : 0 \leq x \leq y \leq z \leq 1 \}}$$，evaluate the following triple integrals.

(a)

$${\iiint_{T} dx dydz}$$

(b)

$${\iiint_{T}e^{x+y+z} dxdydz }$$

2.

Evaluate the following triple integrals

(a)

$${\iiint_{T} dx dydz, \ T = \{(x,y,z):\sqrt{x^2 + y^2} \leq z \leq 3 \}}$$

(b)

$${\iiint_{T}(x^2 + y^2 + z^2) dxdydz, T = \{(x,y,z):\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1\} }$$

3.

Find the centroid of the following regions.．

(a)

，The region is bounded by $y = x$ and $${y = x^2}$$ with the constant density.

(b)

The region is the semisphere $${x^2 + y^2 + z^2 \leq a^2, z \geq 0}$$ with the density proportional to the distance from the center.

(c)

The cone with the radius $a$ and the height $h$ with constant density.

(d)

The region expressed by $${ax \leq x^2 + y^2 \leq a^2}$$.

(e)

The triangular pyramid $T = \{(x,y,z):0 \leq x \leq 1, 0 \leq y \leq 1-x, 0 \leq z \leq 1-x-y\}$. Find $\bar y, \bar z$

(f)

The region is expressed by $${ax \leq x^2 + y^2 \leq a^2}$$with the density is proportional to the distance from the origin.