6. (Exploratory - You'll get full credit for a reasonable effort to express your ideas.) In mathematics, it's very important to devise reasonable definitions. A mathematical definition needs to be precise, capture a particular intuitive idea, and not lead to contradictory statements. In class, you learned how mathematicians define improper integrals of the form / f(x) da and / f(x) dx. In this problem, you'll investigate how one might define an improper integral of the form ( f(x) dx. This problem is exploratory, meaning that you will receive full credit as long as you show evidence of real thought and effort (even if your answers are not completely correct). Ollice we already know how to define both / f(x) de and /f(x) dx, a perfectly reasonable way to define f(x) di would beas ( f(x) dat f(x) dx. (Convince yourself that this is reasonable!) (a) In this part, you'll use this definition to evaluate / x dx. i. What is / x dx? ii. What is / x do? Ill. According to our new definition of / adras / xdx4 / x dx, do you think / x da should converge or diverge? If you think it should converge, what do you think its value should be? (b) In this part, you'll use this definition to evaluate / (x + 3) dr. i. What is [ ( z + 3 ) do ? ii. What is (x+3) da? iii. According to our new definition, do you think (2 + 3) da should converge or diverge? If you think it should converge, what do you think its value should be? (c) Draw pictures showing I de and (7+3) de as signed area. Do you feel that your answers to (a) and (b) make sense with your pictures