# What is the Correct Integral for Finding the Area Below y=0 and Above y=lnx?

• Arman777
In summary, the student is trying to find the area below, above, and to the right of x=0, but is having trouble with the original integral. He finds that the limit as a becomes zero, but is still having trouble with the other term. He is then helped by a friend who points him in the right direction.
Arman777
Gold Member

## Homework Statement

Find the area Below ##y=0##,above ##y=lnx##, and to the right of ##x=0##

## The Attempt at a Solution

I thought an integral like ##\int_0^1 lnx \, dx##
then Its ##-∞## at ##x=0## So I used like ##lim(a→0)=\int_a^1 lnx \, dx## and from that it came
The integral result is ##xlnx-x## so ##1(ln1-1)-a(lna-1)## And if we take limit first term ##1(ln1-1)## is ##-1## but the other term bothers me.It will be ##0(-∞-1)##. I can think like ##lim (a→0)=a ln(a)## and that gave me ##0## but there's also ##+1## so the answer turns ##0## but its impossible.Where I am doing wrong ?

Arman777 said:

## Homework Statement

Find the area Below ##y=0##,above ##y=lnx##, and to the right of ##x=0##

## The Attempt at a Solution

I thought an integral like ##\int_0^1 lnx \, dx##
then Its ##-∞## at ##x=0## So I used like ##lim(a→0)=\int_a^1 lnx \, dx## and from that it came
The integral result is ##xlnx-x## so ##1(ln1-1)-a(lna-1)## And if we take limit first term ##1(ln1-1)## is ##-1## but the other term bothers me.It will be ##0(-∞-1)##. I can think like ##lim (a→0)=a ln(a)## and that gave me ##0## but there's also ##+1## so the answer turns ##0## but its impossible.Where I am doing wrong ?
$$\int_a^1 \ln x \, dx =\left. x \ln x -x \right|_a^1 = 1 \ln 1 - 1 - a \ln a + a$$
What is the limit of that as ##a \to 0+##?

BTW; do not write ##ln x##-- it is ugly and hard to read; instead, write ##\ln x##. You do that by typing "\ln" instead of "ln". (Same for "log", "exp", "lim", "max", "min", all the trig functions and their inverses, and the hyperbolic functions---but not their inverses.)

Which area is described by the given conditions? Draw a picture of it or describe it with words.

Ray Vickson said:
$$\int_a^1 \ln x \, dx =\left. x \ln x -x \right|_a^1 = 1 \ln 1 - 1 - a \ln a + a$$
What is the limit of that as ##a \to 0+##?

BTW; do not write ##ln x##-- it is ugly and hard to read; instead, write ##\ln x##. You do that by typing "\ln" instead of "ln". (Same for "log", "exp", "lim", "max", "min", all the trig functions and their inverses, and the hyperbolic functions---but not their inverses.)
oh ok I foıund thanks

Arman777 said:
oh ok I foıund thanks
I doubt you have found the correct answer since your original integral is wrong. Remember area is$$\int_a^b y_{upper} - y_{lower}~dx$$which is not what you have in your integrand.

LCKurtz said:
I doubt you have found the correct answer since your original integral is wrong. Remember area is$$\int_a^b y_{upper} - y_{lower}~dx$$which is not what you have in your integrand.

Well that make sense...Hmm...ok thanks

## What is an infinite integral area?

An infinite integral area is a concept in calculus that represents the area under a curve that extends infinitely in both the positive and negative directions. It is typically denoted by the symbol ∞ and is calculated using a mathematical technique called integration.

## Why is the concept of infinite integral area important?

The concept of infinite integral area is important because it allows us to calculate the area under a curve that extends infinitely, which is not possible using traditional methods. This is useful in various fields of science and mathematics, such as physics, engineering, and economics.

## How is infinite integral area calculated?

Infinite integral area is calculated using a mathematical technique called integration, which involves breaking down a curve into infinitesimally small sections and summing up their areas. This process is represented by the integral symbol ∫ and is often solved using integration techniques such as substitution or integration by parts.

## What is the difference between a definite and indefinite infinite integral area?

A definite infinite integral area has specific limits of integration, which means the area is calculated within a certain range. On the other hand, an indefinite infinite integral area does not have any limits of integration, and the resulting value is a general solution. In other words, a definite infinite integral area is a specific number, while an indefinite infinite integral area is an expression that can be evaluated for different values.

## What are some real-life applications of infinite integral area?

Infinite integral area has many real-life applications, such as calculating the distance traveled by an object in motion, finding the amount of work done by a force, determining the net change in a quantity over time, and calculating the total revenue or profit in economics. It is also used in fields such as physics, engineering, and statistics to analyze and model various phenomena.

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