class: center, middle, inverse, title-slide .title[ # Multiple Integrals P. II ] .subtitle[ ## Session 10 ] .author[ ### Alejandro Ucan ] .date[ ### 2024-03-31 ] --- # Objetivos: * Interchange orders in double integration. <br/><br/> * Understand some applications of double integrals. <br/><br/> --- # Examle 1: > Find the value of the integral of `\(f(x,y)=4x+2\)` in the bounded region by $ y=x^2$ and `\(y=2x,\)` integrating first with respect to `\(x.\)` --- # Exercise: > Compute the value of the integral of `\(f(x,y)=x\)` in the bounded region by `\(x=0,\)` `\(y=0\)` and `\(y=\sqrt{1-x^2}.\)` --- # Other integral properties: 1. * If `\(f(x,y)\leq 0\)` then `$$\int\int_{R} f(x,y)dA \leq 0.$$` * If `\(f(x,y)\leq g(x,y)\)` in a bounded region `\(R\)` then `$$\int\int_{R}f dA \leq \int\int_R g dA.$$` 1. __Aditivity (in regions):__ Assume that `\(R\)` can be expressed as the union of two disjoint simple regions `\(R_1\)` and `\(R_2,\)` then `$$\int\int_R f dA =\int\int_{R_1}f dA + \int\int_{R_2} f dA.$$` --- # Area of a region: > __Definition:__ the area of a plane region `\(R\)` in `\(\mathbb{R}^2\)` is given by the integral `$$\int\int_{R}dA.$$` __Ejemplo:__ Compute the area of the region bounded by `\(y=\sqrt{x}\)` and `\(y=x^2.\)` --- # Exercise: > Find the area of the region bounded by `\(y=x^2\)` and `\(y=x+2.\)` --- # Mean Value > __Definition:__ the mean value of a function over a region `\(R\)` is defined as the quotient of the integral `$$mean(f,R)=\frac{\int\int_{R}fdA}{\int\int_{R}dA}.$$` __Ejemplo:__ Find the mean value of the function `\(f(x,y)=x\cos(xy)\)` over the rectangle $0\leq x\leq \pi $ and `\(0\leq y\leq 1.\)` --- # Exercise: > The following function describe a distribution of ages in a school where `\(x\)` represent the number of male students and `\(y\)` the number of female students, `\(f(x,y)=xy+x^2+y^3,\)` if we know that `\(0\leq x\leq 10\)` and `\(0\leq y\leq5.\)` Find the average age. --- # Mass and Center of Mass > __Definition:__ Assume that an object occupies a region `\(R\)` in the plane, and the density of the object is given by the function `\(\delta(x,y),\)` then <br/><br/> 1. The mass of the object is given by `$$\int\int_R \delta(x,y)dA.$$` <br/><br/> 1. The mass' moments are: `$$M_x=\int\int_R y\delta(x,y)dA \mbox{ and } M_y=\int\int_R x\delta(x,y)dA.$$` <br/><br/> 1. The mass center is given by `$$\left(\frac{M_y}{M},\frac{M_x}{M}\right).$$` --- # Exercise: > In the triangle bounded by `\(0\leq x\leq 1\)` and `\(0\leq y \leq 2x\)` we have a density distribution by `\(\delta(x,y)=6x+6y+6.\)` Compute the mass and the center of mass. --- # Probability > Assume that we have random variables `\(x\)` and `\(y,\)` and `\(P(x,y)\)` is the probability distribution in a region `\(R,\)` that is, `$$\int_R PdA=1.$$` If we want to know the probability that an event `\(C\)` occurs, then we calculate `$$\int_C PdA.$$` --- #### Example: > Consider the following variables `\(x,y\)` with the probability distribution `$$f(x,y)=\frac{x+2y}{1500}$$` in `\([0,10]\times[0,10].\)` Calculate the probability that happen `\(y\geq 7\)` and `\(x\leq 2\)`.