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Let $ f(x) $ be the probability density function for the lifetime of a manufacturer's highest quality car tire, where $ x $ is measured in miles. Explain the meaning of each integral.

(a) $ \displaystyle \int_{30,000}^{40,000} f(x)\ dx $

(b) $ \displaystyle \int_{25,000}^\infty f(x)\ dx $

(A) $$\int_{30,000}^{40,000} f(x) d x$$

(B) $$\int_{25,000}^{\infty} f(x) d x$$

Applications of Integration

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you have the phone. So it will fix here, is it for our ability, density, then city function off off the life a lifetime on a tire. I'll tell you, we're here. Our arrival x The units are X units, our miles. So we want to give up on interpretation interpretation off the following intervals off this interval internal from family, thousands up to 40 thousands off if fix. Yes. So what these, uh, stands for is the day is the probability that the lifetime, if you call X a lifetime off off the tire satire is between things number 30. Thousands of very case. It is smaller than 40 thousands or Kate. So this is the observation of aviation for 1000? Well, date. So, yeah. That double Meena. Yeah, I probably did that. If you have a tire, the lifetime off that tire is between three. 30 thousands miles and that it is smaller than 40 thousands miles. So in this case, uh, after we careful because here the event, Like in these, uh, in these considerations, the event that, uh, X, the lifetime is equal to say, like, 50 thousands is not included. Mhm. Since we're saying Well, those only upto 40,000 miles, 40,000 miles, 40,000 miles. So you have, uh, in the lifetime off. Your tire, uh, lasts more than 40,000 miles. Well, that event is not included in that probability. On the other is the interval that you like to interpret is, uh is this one, uh, interval from 25 thousands off to infinity, off F effects the X. So this one is the probability probability data. The directs being the lifetime, um, is, uh, is bigger on 25 25 thousands miles. And so since years ago, it goes up to three D. There's no no offer bombs, no upper bounds since the integral, though something community so you're like, is the probability That's the lifetime off the tire on the tire. Uh, okay. He's been here now, Noriko, He's is bigger. Done or or equal to 25 thousands, my ass. So well, in this example here in the event data said that the lifetime is 50,000 miles. This event would fall into this case. Yeah, is considered this event off. The lifetime off the tire lasting 50,000 miles is considered. There it is. It is considered in this the inside of these problems

University of Colorado at Boulder

Applications of Integration