class: center, middle, inverse, title-slide .title[ # Multiple Integrals ] .subtitle[ ## Session 09 ] .author[ ### Alejandro Ucan ] .date[ ### 2024-03-31 ] --- # Goals of the session * Use boxes to approximate the volume under a surface. <br/><br/> * Extend the concept of definite integral to multiple integrals. <br/><br/> * Use Funini's theorem to calculate multiple integrals. <br/><br/> --- # Rectangle partitions and Riemann Sums: Assume that we have a rectangular _region_ `\(R\)` in `\(\mathbb{R}^2,\)` bounded by `\(a\leq x\leq b\)` and `\(c\leq y\leq d.\)` A __partition__ is a _grid_ formed by the pairs of points `\((x_i,y_j)\)` such that `$$x_{i+1}-x_i=\frac{b-a}{n}\quad\mbox{and}\quad y_{j+1}-y_j=\frac{c-d}{m}.$$` --- # Rectangle partitions and Riemann Sums: > If `\(f:\mathbb{R}^2\to \mathbb{R}\)` is a continuous and bounded function in the rectangular region `\(R,\)` then the sum `$$\sum_{i=1}^{n} \sum_{j=1}^m f(x_i,y_j) \Delta x \Delta y$$` _approximates_ the volume under the surface (graph of the function). --- # Refinements of the partition:  --- # The double integral > With better refinements of our partition, our approximation of the volume improves. So, _in the limit_ of refinements we obtain the volume. <br/><br/> The _double integral_ of `\(f(x,y)\)` over the region `\(R\)` is denoted as `\(\int_{R}f\, dA,\)` and is defined as the limit: `$${\int\!\int}_R f(x,y)dA =\lim_{(n,m)\to (0,0)} \sum_{i=1}^{n} \sum_{j=1}^m f(x_i,y_j) \Delta x \Delta y.$$` --- # Properties of the Double integral > __Theorem:__ Any continuous function defined on a rectangle is integrable. <br/> > __Theorem:__ The double integral is linear, that is, if `\(f\)` and `\(g\)` are continuous functions over a region `\(R,\)` then: `$$\int_R (kf+lg)\,dA=k\int_R f\,dA+l\int_R g\,dA.$$` <br/> --- # How to compute double integrals without the sums and the limits? > __Theorem (Fubini):__ If `\(f(x,y)\)` is an integrable function in the region `\(R=[a,b]\times[c,d],\)` then the integral `\(\int_R f\,dA\)` coincides with the quantity: `$$\int_c^d \left[ \int_a^b f(x,y)dx \right]dy=\int_a^b \left[ \int_c^d f(x,y)dy \right]dx.$$` --- # Example 1 Compute the double integral of the function `\(f(x,y)=x^2+yx\)` over the rectangle `\(R=[0,1]\times[0,1].\)` --- # Example 2: Compute the double integral of the function `\(f(x,y)=x^3y-12xy\)` over the rectangle `\(R=[-2,1]\times[0,1].\)` --- # Everything is a rectangle? > __Definition:__ An _elemental region_ in `\(\mathbb{R}^2\)` is a region of the plane that is `\(x-\)`simple and `\(y-\)`simple. <br/> * A region in `\(\mathbb{R}^2\)` is `\(y-\)`simple if it can be written as `$$a\leq x\leq b \quad \mbox{and} \quad \phi_1(x)\leq y \leq \phi_2(x).$$` <br/><br/> * A region in `\(\mathbb{R}^2\)` is `\(x-\)`simple if it can be written as `$$c \leq y \leq d \quad \mbox{and} \quad \psi_1(y)\leq x\leq \psi_2(y).$$` --- # General Statement of Fubini's theorem: > __Theorem:__ If `\(D\)` is a region simple and `\(f\)` is integrable in `\(D,\)` then `$$\int_D f\,dA = \int_a^b \int_{\phi_1(x)}^{\phi_2(x)} f(x,y)dydx.$$` --- # Example 3: Compute the value of the integral of `\(f(x,y)=\sqrt{\frac{x}{y}}\)` in the region `\(R\)` bounded by `\(0\leq x \leq 1\)` y `\(x^2\leq y \leq x.\)` --- # Ejemplo 4: Find the volume of the prism whose base is the region in the plane `\(xy\)` bounded by the lines `\(y=x\)` and `\(y=2x,\)` and whose top is the plane `\(z=3-x-y.\)` --- # Example 5: Find the volume of the regio bounded by the functions `\(f(x,y)=xy\)` and the function `\(f(x,y)=16-x^2-y^2,\)` over the region `\(R\)` bounded by the curves `\(y=2\sqrt{x},\)` `\(y=4x-2\)` and the `\(x-\)`axis.