Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Suppose that $f$ and $g$ are integrable on $[a, b],$ but neither$f(x) \geq g(x)$ nor $g(x) \geq f(x)$ holds for all $x$ in $[a, b]$[i.e., the curves $y=f(x)$ and $y=g(x)$ are intertwined].(a) What is the geometric significance of the integral$$\int_{a}^{b}[f(x)-g(x)] d x ?$$$$\text { (b) What is the geometric significance of the integral }$$$$\int_{a}^{b}|f(x)-g(x)| d x ?$$

(a) (area above graph of $g$ and below graph of $f$ ) minus (area abovegraph of $f$ and below graph of $g$ )(b) area between graphs of $f$ and $g$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

00:45

calculate the integral, as…

08:21

(a) Use integration by par…

04:09

Show that if $f(x)$ is int…

00:38

Give a geometrical explana…

01:43

02:27

Suppose that$$f(x)=\fr…

00:20

05:48

02:40

Think About It When is…

06:26

Product of integrals Suppo…

eso basically, in this both these problems with a, um since the functions air intertwined, they must intersect it at least one point, um, Simon and draw a couple points of intersection here and say, this is a MB. Um, any time you have, um, like, one function being above the other, you either get a positive or a negative area, but then I'm trying to color code it. Then the after they intersect, there's gonna be a portion that crosses it out. They, um, cancel each other out. Um, so what basically happens is part A, since there's no absolute value, Um, there's going to be portions that cancel out. Um, And if you were to actually subtract the values from each other and you drew a new graf, depending, if he didn't like the red before the blue, you know, you'd have an answer that looks something like this. And the way I drew it ended up being like a negative area. But whatever. Um and so you above the X axis and below the X axis would cancel each other out so the areas cancel. So what? What important thing doesn't do when you put the absolute value in there. So let's say it's the same graph. Um, from A to B is drawing a new one for you than everywhere. It dips below the X axis. The absolute value would make it positive, something that, um so in part B, when you get to absolute value now you actually get the area between the curves. And that's the important thing about this problem is that you understand that premise.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

calculate the integral, assuming that $f$ is integrable and $\int_{1}^{b} f(…

(a) Use integration by parts to show that$$ \int f(x) dx = xf (x) - \int…

Show that if $f(x)$ is integrable on every interval of real numbers, and if …

Give a geometrical explanation of why $\int_{a}^{a} f(x) d x=0$

Give a geometrical explanation of why $\int_{a}^{a} f(x) d x=0$.

Suppose that$$f(x)=\frac{d}{d x}(1-\sqrt{x}) \quad \text { and } \quad g…

Suppose that$$f(x)=\frac{d}{d x}(1-\sqrt{x}) \text { and } g(x)=\frac{d}…

Think About It When is$$\int_{a}^{b} f(x) d x=\int_{a}^{b}|f(x)| d x ?$$…

Product of integrals Suppose $f(x, y)=g(x) h(y),$ where $g$ and $h$ are cont…

06:18

Express the integral in terms of the variable $u,$ but do not evaluate it.

00:53

Use a graphing utility to generate the graph of$$y=\left(1+\frac{1}{…

02:00

Determine whether the statement is true or false. Explain your answer.Th…

01:19

Find the exact arc length of the curve over the interval.$$24 x y=y^…

02:18

Suppose that a tumor grows at the rate of $r(t)=k t$ grams per week for some…

03:02

The flat surfaces shown are submerged vertically in water. Find the fluid fo…

00:48

A variable force $F(x)$ in the positive $x$ -direction is graphed in the acc…

01:49

Consider the curve $y=x^{2 / 3}$(a) Sketch the portion of the curve betw…

02:06

Use a graphing utility, where helpful, to find the area of the region enclos…

01:46

Let $y=f(x)$ be a smooth curve on the interval $[a, b]$ andassume that $…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.