Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Suppose that $f$ is a nonnegative-valued, continuous function defined on $R=\{(x, y) \mid a \leq x \leq b, c \leq$ $y \leq d\}$. If $f(x, y) \leq M$ for some positive number $M$ explain why the volume $V$ under the graph of $f$ over $R$ is at most $M(b-a)(d-c)$ .

Calculus 3

Chapter 5

Multiple Integration

Section 1

Introduction: Areas and Volumes

Multiple Integrals

Johns Hopkins University

Missouri State University

Baylor University

Boston College

Lectures

04:18

In mathematics, an integra…

26:18

In mathematics, a double i…

02:31

Suppose $f(x)$ is positive…

01:35

Suppose $f$ and $g$ are co…

03:00

Explain why, if $f$ is con…

01:17

01:43

02:11

07:36

Let $f$ and $g$ be two con…

00:59

Use the graphical interpre…

04:34

01:56

Suppose $ f $ is a continu…

01:30

Assume $f(x)$ is continuou…

07:26

If $f(x, y)$ is continuous…

00:40

04:02

Suppose that $ f $ and $ g…

01:41

Suppose that $f$ is contin…

01:11

Let $g(x)=-f(x)$ where $f$…

01:14

03:56

The graph of the derivativ…

04:20

05:16

Explain why, if $f(a) \geq…

So, let's look at the integral f of a 2 f of b, f inverse y d. U i will use the substitution f. Inverse y is equal to? U. This gives f. F. U is equal to y taking derivatives. We get f! Prime. U d! U is substituting a to b? U, prime? U? D? U now, integrating by parts this becomes. This is equal to b, f of b minus a of a minus a to b, f of x dx. We change u 2 x, because, u is just a very well! So this gives us a to be f x. D x plus, is equal to b of b minus a f f a.

View More Answers From This Book

Find Another Textbook

In mathematics, an integral assigns numbers to functions in a way that can d…

In mathematics, a double integral is an integral where the integrand is a fu…

Suppose $f(x)$ is positive, continuous, and increasing over the interval $[a…

Suppose $f$ and $g$ are continuous functions such that $\int_{c}^{d} f(t) d …

Explain why, if $f$ is continuous over $[a, b],$ there is at least one point…

Explain why, if $f$ is continuous over $[a, b]$ , there is at least one poin…

Explain why, if $f$ is continuous over $[a, b]$ and is not equal to a consta…

Let $f$ and $g$ be two continuous functions such that $0 \leq m_{1} \leq f(x…

Use the graphical interpretation of the definite integral to explain the ine…

Suppose $ f $ is a continuous function where $ f(x) > 0 $ for all $ x $, …

Assume $f(x)$ is continuous for $g \leq x \leq c \leq c$ as shown in the fig…

If $f(x, y)$ is continuous over $R: a \leq x \leq b, c \leq y \leq d$ and

Assume $f(x)$ is continuous for $g \leq x \leq c$ as shown in the figure. Wr…

Suppose that $ f $ and $ g $ are continuous on $ [a, b] $ and differentiable…

Suppose that $f$ is continuous and positive-valued everywhere and that the $…

Let $g(x)=-f(x)$ where $f$ has non-negative function values on an interval $…

Suppose that $f$ is continuous on the interval $[0,1]$ and that $0 \leq f(x)…

The graph of the derivative $f^{\prime}$ of a continuous function $f$ is sho…

Explain why, if $f(a) \geq 0$ and $f$ is increasing on $[a, b],$ that the le…