9. NORMALIZING A GAUSSIAN The function f(x) = e ' comes up in many applications, from fitting the distribution of scores on an exam, to the fluorescent signal of a single molecule, to the distribution of velocities of molecules in a gas. Suppose we want to normalize a distribution of this form, that is, we want to find N such that becomes large. We We don't know how to deal with the infinite limits, but er becomes very small as will therefore approximate a.) Write a sum, using sigma notation, that approximates the integral Leerder using 10 rectangles of equal width, evaluating the function on the right side of the interval. - dr using 10 rectangles of b.) Write a sum, using sigma notation, that approximates the integral equal width, using the trapezoid rule. c.) Use the following code to evaluate the integral L and with Matlab's integral function. e dr using the two methods with 200 intervals, Rectangles x_min=-5; x_max=5; n-200; dx=(x_max-x_min); xx_min+dx:x:x_max f-exp(-x. 2); I-sum(fdx) Trapezoid x_min=-5; x_max-5; n-200; dx=(x_max-x_min); x1-x_min+dxdx:x_Bax; f1-exp(-x.2); x2-x_min:dx:x_max-dx; f2-exp(-x.^2); I=(1/2)*(sum(f1*dx) +sum (f2+dx)) Matlab I=integral (@(x) (exp(-x.^2)),-5,5) T. How d.) For the particular integral e do it is possible to do a clever trick to find a solution: good are your approximations? (Calculate the percent error, which is Percent error = 100. exact value) - (approximation) (exact value) d.) What is the normalization constant N