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$)$

$50,000$f$($x$)$ dx$30,000$

The integral is the probability that a randomly chosen tire will have a lifetime between $30,000$ and $50,000$ miles.The integral is the probability that a randomly chosen tire will have a lifetime under $50,000$ miles.$$The integral is the probability that a randomly chosen tire will have a lifetime of at least $30,000$ miles.The integral is the probability that a randomly chosen tire will have a lifetime of exactly $50,000$ miles.

$($b$)$

f$($x$)$ dx$25,000$

The integral is the probability that a randomly chosen tire will have a lifetime of at least $25,000$ miles.The integral is the probability that a randomly chosen tire will have a lifetime under $25,000$ miles.$$The integral is the probability that a randomly chosen tire will not wear out.The integral is the probability that a randomly chosen tire will have a lifetime of exactly $25,000$ miles.