signed area under the curve between 0 and x Note that on the

Last updated: 8/17/2023

signed area under the curve between 0 and x Note that on the left hand side of the the variable is x Ther on the right hand side is a dummy variable that represents all the values that x may take You can call this dummy variable anything you like without changing f x AS LONG AS you don t call the dummy variable x again For example F x f 1 dt f u du ff s ds You should NOT write F x f x dx 1 Let y f x x Then consider the signed area function F x f t dt which you know represents the signed area below the curve from 0 to x For example F 2 represents the area below the curve y f from 0 to 2 as show below A Using your graphing calculator to numerically approximate the definite integrals complete the data chart below x F x f 1 dt I 2 3 4 B Now you ll use your calculator to create an area function F x By area function we mean a function where you enter an x value and the output is the area under the curve f x x The function will be in the form axb and will represent the area function F x You can do this by doing a regression on your calculator enter your data points from the table you filled in and your calculator will try to find a function that matches those data points On the TI 83 this done by pressing STATIEDIT and then entering your data into the statistical register x values go in L1 F x values go in L2 Then go STATICALCJA PwrReg which will find a function in the form y ax that fits your data points See pages 12 27 in the TI 83 book if needed Remember that the calculator is approximating so if it tells you that a 122 99999999 and b 12 00000001 it s legitimate to just take a 123 and b 12 Your function approximating F x is C What is the relationship between your function approximating F x and the antiderivative of f x x how are they similar and how are they different D Using the relationship you just described above and taking x 4x3 12x2 2x take a guess at an equation for the function F x f t dt the signed area function for f In other words take a quess at an area function that will give the area under the curve f x 4x 12x 2x Your guess F x f 1 dt E Now using your formula for F x your guess in part D and your calculator to numerically approximate the definite integrals complete the table below and discuss whether your formula for F x seems to work f dt calculator approximation Your F x r 2 3 F Based on your work can you draw any general conclusions concerning functions f x and