class: center, middle, inverse, title-slide .title[ # Integration Techniques and Applications ] .subtitle[ ## Session 03 ] .author[ ### Alejandro Ucan ] .date[ ### 2023-11-12 ] --- # Goals: * Substitution. <br/><br/> * By parts. <br/><br/> * Center of Mass. <br/><br/> --- # Substitution > __Method:__ Substitution is a method of integration in which we use a change of variable to transform the integrand into a form that is easier to integrate (and use direct integration). <br/><br/> `$$\int_{a}^{b} f(g(x))g'(x)dx = \int_{g(a)}^{g(b)} f(u)du$$` -- #### Example 1. `\(\int_{0}^{1} \frac{2x+2}{(x^2 + 2x + 1)} dx\)` <br/><br/> 2. `\(\int_{-1}^{1} xe^{3x^2} dx\)` <br/><br/> 3. `\(\int_{0}^{\pi/4} \tan(x)dx\)` <br/><br/> --- # By Parts > __Method:__ By parts is a method of integration in which we use the product rule of differentiation to transform the integrand into a form that is easier to integrate (and use direct integration). <br/><br/> `$$\int_{a}^{b} f(x)g'(x)dx = f(x)g(x) \Big|_{a}^{b} - \int_{a}^{b} f'(x)g(x)dx$$` __Remarks:__ The choice of `\(f\)` and `\(g'\)` in the integral depend in the following criteria: the choice of `\(f\)` must be the function that simplify when we derive it, and the choice of `\(g'\)` should be the one that is easier to integrate. <br/><br/> --- #### Example 1. `\(\int_{0}^{1} x\ln(x)dx\)` <br/><br/> 2. `\(\int_{0}^{1} xe^x dx\)` <br/><br/> 3. `\(\int_{0}^{1} x\cos(x) dx\)` <br/><br/> --- # Center of Mass > __Definition:__ Given a wire with endpoint `\((a,0)\)` and `\((b,0)\)`, that has density `\(\rho(x)\)`, the center of mass of the wire is given by the following formula: <br/><br/> `$$\overline{x}=\displaystyle{\frac{\int_{a}^{b} \rho(x)x dx}{\int_a^b \rho(x)dx}}$$` and the point where is the center of mass is given by `\((\overline{x},0).\)` --- #### Example > Consider a wire from `\((0,0)\)` to `\((10,0)\)` with density function given by `\(x.\)` Find the center of mass. -- <br/><br/> #### Solution: 1. We compute the corresponding integrals: `$$\int_{0}^{10} x dx = 50$$` and `$$\int_{0}^{10} x^2 dx = \frac{1000}{3}$$` <br/><br/> 2. We compute the center of mass: `$$\overline{x} = \frac{\frac{1000}{3}}{50} = \frac{20}{3}$$` <br/><br/> 3. The center of mass is given by `\((\frac{20}{3},0).\)` --- #### Example > Consider a wire from `\((0,0)\)` to `\((\pi,0)\)` with density function given by `\(\sin(x).\)` Find the center of mass. -- <br/><br/> #### Solution: 1. We compute the corresponding integrals: `$$\int_{0}^{\pi} \sin(x) dx = 2$$` and `$$\int_{0}^{\pi} x\sin(x) dx = \pi$$` <br/><br/> 2. We compute the center of mass: `$$\overline{x} = \frac{\pi}{2}$$` <br/><br/> 3. The center of mass is given by `\((\frac{\pi}{2},0).\)`