class: center, middle, inverse, title-slide .title[ # Triple Integrals ] .subtitle[ ## Session 12 ] .author[ ### Alejandro Ucan ] .date[ ### 2024-04-14 ] --- # Goals for the session: * Know the volume differential. <br/><br/> * Define the integral in `\(\mathbb{R}^3\)`. <br/><br/> * Calculate triple integrals in elementary regions. --- # Triple Integral: > The notion of triple integral is a generalization of the concept of integral to three-dimensional space. Using this, we can calculate the "volume" of a solid of four dimensions. The naive idea comes from the volume of a box, in fourth dimension, the box will have 4 different sides, and the volume will be the product of the four sides. Therefore in our case, `$$f(x,y,z)dx dy dz$$` is the volume of a box-base with sides `\(dx, dy\)` and `\(dz\)`. --- # How to compute triple integrals? Fubini's theorem is valid for triple integrals, so to calculate it, we just need to perform an iterated integration. `$${\int\!\int\!\int}_B fdV=\int_a^b \int_c^d \int_e^f f(x,y,z)dzdydx.$$` Recall that in the case of numerical limits, the change does not affect the result. --- # Triple integral in elementary regions: > **Definition:** an *elementary* region in `\(\mathbb{R}^3\)` is a region in which one of the variables is bounded between two functions of the other two variables. <br/><br/> **Example:** The *ball* of radius one in `\(\mathbb{R}^3\)` is an elementary region described by the following inequalities: `$$-1\leq x\leq 1,$$` `$$-\sqrt{1-x^2} \leq y\leq \sqrt{1-x^2},$$` $$ -\sqrt{1-x^2-y^2} \leq z \leq \sqrt{1-x^2-y^2}.$$ --- # Other Examples: Other examples of elementary regions are: * Consider the following cylinder: `$$-1\leq x \leq 1$$` `$$-\sqrt{1-x^2}\leq y \leq \sqrt{1-x^2}$$` $$ -2\leq z \leq 2.$$ * Consider the following prism: `$$0 \leq x \leq 2$$` `$$0\leq y \leq 2$$` `$$0\leq z\leq 5-2x-2y$$` --- # Example 1: > Find the volume of the solid enclosed by the function `\(f(x,y,z)=z\)` over the region bounded by `\(x=0,\,y=0,\, z=0\)` y `\(x+y+z=1\)` -- `$$R: 0\leq x\leq 1,\, 0\leq y \leq 1-x,\, 0\leq z\leq 1-x-y.$$` `$$\int\int\int_R z dV= \int_0^1\int_0^{1-x}\int_0^{1-x-y} z dzdydx$$` --- # Example 2: > Find the volume of the solid enclosed by the function `\(f(x,y,z)=z\)` over the region bounde by the paraboloid `\(y=x^2+z^2\)` and `\(y=4.\)` -- `$$E: -2\leq x\leq 2,\, -\sqrt{4-x^2}\leq z\leq \sqrt{4-x^2},\, x^2+z^2\leq y\leq 4$$` `$$\int\int\int_E \sqrt{x^2+z^2}dV= \int_{-2}^2\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{x^2+z^2}^4 z dydzdx$$` --- What if we try with the following limits: `$$0\leq y\leq4,\, -\sqrt{y}\leq x \leq \sqrt{y},\, -\sqrt{y-x^2}\leq z \leq \sqrt{y-x^2}$$` --- #### Exercise 1: Compute the following triple integrals: * `\(\int_0^1\int_0^z\int_0^{x+z}6xzdydxdz\)` * `\(\int_0^1\int_0^z\int_0^y ze^{-y^2}dxdydz\)` * `\(\int_0^{\sqrt{\pi}}\int_0^x\int_0^{xz}x^2\sin(y)dydzdx\)` --- #### Exercise 2: Draw the region `\(E\)` and compute the integral of `\(f(x,y,z)=6.\)` The region is given under the plane `\(z=1+x+y\)` and above `\(z=0\)` bounded by the curves `\(y=\sqrt{x},\, y=0\)` y `\(x=1.\)`