Chapter 5

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{3} \frac{\ln x}{x} d x \text { Hint }: \text { Let } u=\ln x $$

Let \(f\) be continuous on \([a, b]\) and thus integrable there. Show that $$ \left|\int_{a}^{b} f(x) d x\right| \leq \int_{a}^{b}|f(x)| d x $$

Suppose that \(f^{\prime}\) is integrable and \(\left|f^{\prime}(x)\right| \leq M\) for all \(x\). Prove that \(|f(x)| \leq|f(a)|+M|x-a|\) for every \(a\).

. Water leaks out of a 200-gallon storage tank (initially full) at the rate \(V^{\prime}(t)=20-t\), where \(t\) is measured in hours and \(V\) in gallons. How much water leaked out between 10 and 20 hours? How long will it take the tank to drain completely?

Oil is leaking at the rate of \(V^{\prime}(t)=1-t / 110\) from a storage tank that is initially full of 55 gallons. How much leaks out during the first hour? During the tenth hour? How long until the entire tank is drained?

The mass, in kilograms, of a rod measured from the left endpoint to the point \(x\) meters away is \(m(x)=x+x^{2} / 8\). What is the density \(\delta(x)\) of the rod, measured in kilograms per meter? Assuming that the rod is 2 meters long, express the total mass of the rod in terms of its density.

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{3 i}{n}\right)^{2} \frac{3}{n} $$

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{2 i}{n}\right)^{3} \frac{2}{n} $$

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[\sin \left(\frac{\pi i}{n}\right)\right] \frac{\pi}{n} $$

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[1+\frac{2 i}{n}+\left(\frac{2 i}{n}\right)^{2}\right] \frac{2}{n} $$