I need help in calculus 2

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[spring slate]

i need help in calculus 2 can anyone help me figure it out?

[random dagger]

https://www.integral-calculator.com/

you can input your integral into any online CAS

btw that's neither computer science nor linear algebra question

[spring slate]

it doesnt let me put calculus tag on

[random dagger]

real analysis is a more appropriate tag

[spring slate]

i did this but it doesn t explain how it applies fundamental theorem of calculus

[random dagger]

you can click the "show steps" button to generate the steps

[orchid estuary]

By definition, $\int_1^\infty\frac{1}{x^2}\dd{x}:=\lim_{a\to\infty}\int_1^a\frac{1}{x^2}\dd{x}$

[tacit phoenix]

Omegabet_

[orchid estuary]

you know how to compute $\int_1^a\frac{1}{x^2}\dd{x}$ via FTC, then just take the limit

[tacit phoenix]

Omegabet_

[spring slate]

if i apply FTC its -1/2x^2?

[orchid estuary]

no

vin posted stuff that isnt relevant lol

what's a primitive/anti-derivative of 1/x^2 = x^(-2)?

[spring slate]

X^-1/2?

[orchid estuary]

clearly not

$\int x^n\dd{x}=\frac{x^{n+1}}{n+1}+C$

[tacit phoenix]

Omegabet_

[orchid estuary]

so what is $\int x^{-2}\dd{x}$?

[tacit phoenix]

Omegabet_

[spring slate]

x^-1/-1?

[orchid estuary]

+C but yes, -1/x + C

so what, by FTC2, is $\int_1^a\frac{1}{x^2}\dd{x}$?

[tacit phoenix]

Omegabet_

[spring slate]

so this is 1x^-2?

[orchid estuary]

.....

[spring slate]

and anti derevative would be

[orchid estuary]

we already discussed what the anti-derivative is....

[orchid estuary]

.

remember a minute ago?

[spring slate]

so i put a and 1 in x?

according to FTC@

2

F(b) - F(a)

[orchid estuary]

you use verbatim what FTC2 says

yes, and F(x) is clearly -1/x

[spring slate]

-1/a - -1/1 so answer will be -1/a

[orchid estuary]

what's 1/1...?

[spring slate]

-1/+∞

[orchid estuary]

what's 1/1...?

[spring slate]

1

[orchid estuary]

so what's -1/a - -1/1?

[spring slate]

-1

[orchid estuary]

????

[spring slate]

1/a -1

[orchid estuary]

go back to grade 4 and recall algebra

what is -1/a - -1/1?

hint: minus signs dont magically leave.

[spring slate]

-1/a + 1/

[orchid estuary]

yes, review your algebra if you're making mistakes like that

$\int_1^a\frac{1}{x^2}\dd{x}=-[\frac{1}{x}]_1^a=-[\frac{1}{a}-\frac{1}{1}]=1-\frac{1}{a}$

[tacit phoenix]

Omegabet_

[spring slate]

i see now

[orchid estuary]

and as a goes to infinity...?

[spring slate]

its a limit where a->+∞

[orchid estuary]

congrats on reading

now evaluate that limit

without saying nonsense like 1/inf

[spring slate]

it would be 1

@orchid estuary

[orchid estuary]

yes

[spring slate]

That’s the answer?

[orchid estuary]

duh?

[orchid estuary]

$\int_1^\infty\frac{1}{x^2}\dd{x}=\lim_{a\to\infty}\int_1^a\frac{1}{x^2}\dd{x}=\lim_{a\to\infty}1-\frac{1}{a}=1$

[tacit phoenix]

Omegabet_

[spring slate]

i got it thanks so much

Is this correct?

@orchid estuary

[orchid estuary]

looks it

$\dd{V}=A(x)\dd{x}=\pi r(x)^2\dd{x}=\pi(\sqrt{x-1}-0)^2\dd{x}=\pi(x-1)\dd{x}$

[tacit phoenix]

Omegabet_

[spring slate]

@orchid estuary this one?

did i apply FTC correctly?

[orchid estuary]

if you're going to just ping me to check work, don't.

any calculator like wolfram can check your computations for you.

[spring slate]

i need some guidance.. i didnt know how to do this and u tought me and i wanna see if i got it

[orchid estuary]

then use a calculator to check

you're pinging me to check your work, not help you.

[spring slate]

on this one?

[orchid estuary]

besides, you shouldn't be pinging people in the 1st place

[spring slate]

ok sorry

[spring slate]

i have no clue how to do this do u have time to explain? if no its cool

[orchid estuary]

draw the region, then use the formula.

[spring slate]

which formula

[orchid estuary]

well let's use common sense here

the question asks about area of a region

and that region is bounded by 2 curves

so my guess is, the formula for area between two curves

[spring slate]

∫f(x) - g(x) dx

?

[orchid estuary]

yep.

[spring slate]

I sketched them now how do I use that formula?

[spring slate]

+close