Next number in a sequence (definite integral problem)

[5ymmetrica1]

(definite integral problem) write as an expression

Homework Statement

\int₀^k 3x²
suggest an expression for definite integrals from x=0 to x=k

Homework Equations

see below

The Attempt at a Solution

when k= 0, \int₀^k 3x² = 0
when k= 1, \int₀^k 3x² = 1
when k= 2, \int₀^k 3x² = 8
when k= 3, \int₀^k 3x² = 27
when k= 4, \int₀^k 3x² = 64
when k= 5, \int₀^k 3x² = 125
when k= 6, \int₀^k 3x² = 216

but now the question is asking me to find an expression for when k = k in \int₀^k 3x²
I'm not sure how to go about this, can someone point me in the right direction please?

 

[Curious3141]

Homework Statement

\int₀^k 3x²
suggest an expression for definite integrals from x=0 to x=k

Homework Equations

see below

The Attempt at a Solution

when k= 0, \int₀^k 3x² = 0
when k= 1, \int₀^k 3x² = 1
when k= 2, \int₀^k 3x² = 8
when k= 3, \int₀^k 3x² = 27
when k= 4, \int₀^k 3x² = 64
when k= 5, \int₀^k 3x² = 125
when k= 6, \int₀^k 3x² = 216

but now the question is asking me to find an expression for when k = k in \int₀^k 3x²
I'm not sure how to go about this, can someone point me in the right direction please?

I'm assuming you haven't actually learned how to do integration yet?

In that case, does a pattern jump out at you in these numbers: 0, 1, 8, 27, 64, ...?

Hint: think of a BOX.

 

[5ymmetrica1]

thanks for the reply,
no I am actually mid-way through my course on integration, and have been going fine until this problem.

Ive tried a bunch of patterns
such as...
0= 0x0, 1=1x1, 8=2x4, 27=3x9, 64=4x16, 125=5x25, 216= 6x36

but all I can conclude from this is that the number is the sum of its number in the sequence and the square of that number
(i.e 27 is the 3rd number in the sequence and 3² is 9, therefore 27 = 3x9)

but I'm not sure how to write this as an expression for the upper limit 'k'

can I say \int₀^k 3x² dx = k(k²)

 

[SithsNGiggles]

What's the product of a number and its square equivalent to? Like Curious said, "think of a BOX."

 

[5ymmetrica1]

Im not sure what you mean by that.
If its a box, then 3x9 is the dimensions of the box and 27 is the area.

But how does this help me write an expression for
\int₀^k 3x² dx

 

[Curious3141]

Im not sure what you mean by that.
If its a box, then 3x9 is the dimensions of the box and 27 is the area.

But how does this help me write an expression for
\int₀^k 3x² dx

A more explicit hint, then:

x*x = x SQUARED (x²)

x*x*x = ? (x^?)

 

[5ymmetrica1]

so you mean \int₀^k 3x² dx = k³

 

[5ymmetrica1]

rather than starting another post I also have two more sets of these problems which I have been struggling with finding the pattern.

for 3x²

when k= -1, ∫ _-1^k 3x² = 0
when k= 0, ∫ _-1^k 3x² = 1
when k= 1, ∫ _-1^k 3x² = 2
when k= 2, ∫ _-1^k 3x² = 9
when k= 3, ∫ _-1^k 3x² = 28
when k= 4, ∫ _-1^k 3x² = 65
when k= 5, ∫ _-1^k 3x² = 126
when k= k, ∫ _-1^k 3x² = ??

so the sequence is [0,1,2,9,28,65,126...]
I believe the answer for this is (k)³ +1

and also for

when k= -2, ∫ _-2^k 3x² = 0
when k= -1, ∫ _-2^k 3x² = 7
when k= 0, ∫ _-2^k 3x² = 8
when k= 1, ∫ _-2^k 3x² = 9
when k= 2, ∫ _-2^k 3x² = 16
when k= 3, ∫ _-2^k 3x² = 35
when k= 4, ∫ _-2^k 3x² = 72
when k= k, ∫ _-2^k 3x² = ??

so the sequence is [0,7,8,9,16,35,72...]

I need to find the expression for 'k' in each of these but I can't find any pattern in the numbers when applying the same technique as I did in the first part of my question.

 

[Curious3141]

so you mean \int₀^k 3x² dx = k³

Right.

 

[Curious3141]

rather than starting another post I also have two more sets of these problems which I have been struggling with finding the pattern.

for 3x²

when k= -1, ∫ _-1^k 3x² = 0
when k= 0, ∫ _-1^k 3x² = 1
when k= 1, ∫ _-1^k 3x² = 2
when k= 2, ∫ _-1^k 3x² = 9
when k= 3, ∫ _-1^k 3x² = 28
when k= 4, ∫ _-1^k 3x² = 65
when k= 5, ∫ _-1^k 3x² = 126
when k= k, ∫ _-1^k 3x² = ??

so the sequence is [0,1,2,9,28,65,126...]
I believe the answer for this is (k)³ +1

Right also.

Another way to write it is $k^3 + 1^3$, correct? The cube of 1 is of course also 1. I'm writing it this way to give you a big hint for the next part.

and also for

when k= -2, ∫ _-2^k 3x² = 0
when k= -1, ∫ _-2^k 3x² = 7
when k= 0, ∫ _-2^k 3x² = 8
when k= 1, ∫ _-2^k 3x² = 9
when k= 2, ∫ _-2^k 3x² = 16
when k= 3, ∫ _-2^k 3x² = 35
when k= 4, ∫ _-2^k 3x² = 72
when k= k, ∫ _-2^k 3x² = ??

so the sequence is [0,7,8,9,16,35,65,72...]

I need to find the expression for 'k' in each of these but I can't find any pattern in the numbers when applying the same technique as I did in the first part of my question.

Where did that "65" (in bold) come from? It shouldn't be there, should it?

From the way I wrote the last expression, can you now figure this one out? It's all got to do with cubes, and it's a simple pattern.

 

[5ymmetrica1]

oh woops the 65 is from copying the first section so disregard that.

thanks for the help curious, I'm getting the result for the last one of k³ + 2³ does that look right to you?

 

[Curious3141]

oh woops the 65 is from copying the first section so disregard that.

thanks for the help curious, I'm getting the result for the last one of k³ + 2³ does that look right to you?

Yes, correct, altho' you can of course write it more simply as $k^3 + 8$.

 

[5ymmetrica1]

thanks again curious!

I just have one last question.
Im asked to consider the function y=4x³+3x² and use my findings from the first part of this post to suggest a function that will generate the definite integral for y=4x³+3x², from x=a to x=k

then I am asked to write an integral expression for the same function, which I can do
\int_a^k4x³+3x² = \int_a^k4x³ + \int_a^k3x²

but I'm not sure about the part asking me to use the information I've found to create a function that will generate the definite integral. Could you possibly give me any clues into how to go about this question?

 

[Curious3141]

thanks again curious!

I just have one last question.
Im asked to consider the function y=4x³+3x² and use my findings from the first part of this post to suggest a function that will generate the definite integral for y=4x³+3x², from x=a to x=k

then I am asked to write an integral expression for the same function, which I can do
\int_a^k4x³+3x² = \int_a^k4x³ + \int_a^k3x²

but I'm not sure about the part asking me to use the information I've found to create a function that will generate the definite integral. Could you possibly give me any clues into how to go about this question?

OK, you know how to split the integral up.

Let's tackle the second part because the integrand (the expression you're integrating) is the same as what you worked on before. Only the lower bound is different.

When your lower bound was -1, the expression was $k^3 + 1^3$, also expressible as $k^3 - (-1)^3$.

When your lower bound was -2, the expression was $k^3 + 2^3$, also expressible as $k^3 - (-2)^3$.

Can you suggest what the expression might be for a lower bound of a?

For the first integral, you know the form has to be similar to this. But note that the integrand has a one higher exponent (power) for x now. So maybe it follows a similar pattern, just with one power higher? Try checking.

Just curious (that's my name, after all ) about this course you're taking - are they doing this prior to teaching you how to do the actual integration? Because it seems a very weird way of introducing the topic, in my opinion. Do you even have a conceptual understanding of what it means to take a definite integral? An indefinite one? Have you covered differentiation, even?

Anyway, notice that the integrands are always of the form $nx^{n-1}$, which always has the integral $x^n$. Actually, it's $x^n + c$, where c is an arbitrary constant, but this cancels out when you're doing definite integrals. Hopefully, you'll cover all this rigorously soon, because I can't, for the life of me, see why they're teaching your course this way.

 

[5ymmetrica1]

so would the expression be k³ - a³

and for 4x³

I'm getting these results

\int₀^k4x³dx = k⁴

\int_-1^k4x³dx = k⁴ - 1

\int_-2^k4x³dx = k⁴ - 2⁴

so if my first expression is right would this be k⁴ - a⁴ for this one?and to answer your question about integration, so far I've been taught about upper and lower sums, definite integrals and yes we've done differentiation, such as the product and chain rule, implicit differention and finding differentials from the first principles. But I agree it is a strange order to teach calculus in.

 

[Curious3141]

so would the expression be k³ + a³

No! Careful with the sign. Think about it, when a = -1, does your expression give you the expected result?

and for 4x³

I'm getting these results

\int₀^k4x³dx = k⁴

\int_-1^k4x³dx = k⁴ - 1

\int_-2^k4x³dx = k⁴ - 2⁴

so if my first expression is right would this be k⁴ - a⁴ for this one?

This is correct, so your wrong sign in the expression above may have just been a typo. On the other hand, you may be missing the complete picture here. Remember (for example), $k^4 - 2^4$ is equivalent to $k^4 - (-2)^4$. You should be able to see how this ties into the earlier expression.

A bit of background may help: with definite integration, the last step always involves a subtraction. Evaluate the final expression at the upper bound and the lower bound. Then take the former minus the latter. It's always subtraction, not addition, so this should give you a big clue.

and to answer your question about integration, so far I've been taught about upper and lower sums, definite integrals and yes we've done differentiation, such as the product and chain rule, implicit differention and finding differentials from the first principles. But I agree it is a strange order to teach calculus in.

Well, I guess your instructors know best what they're planning.

 

[5ymmetrica1]

yes it was a typo, thanks again for your excellent help curious!

 

[Curious3141]

yes it was a typo, thanks again for your excellent help curious!

Suggest you read the stuff I added in my edit.

 

[5ymmetrica1]

have gone through your edit and tried to implement it into my expression.
can I say that \int_a^k 4x³+3x²dx = (k⁴ - a⁴) - (k³ - a³)

or am I still misunderstanding something?

 

[Curious3141]

have gone through your edit and tried to implement it into my expression.
can I say that \int_a^k 4x³+3x²dx = (k⁴ - a⁴) - (k³ - a³)

or am I still misunderstanding something?

Each of the "split" integrals is correct, i.e. $k^4 - a^4$ and $k^3 - a^3$.

My question is: why are you subtracting one of those from the other, when you should be adding?

What I said (subtract lower bound value from upper bound value) applies to a single integral. You'd correctly split the integrals into two, so you should just work out each individual integral (which you did correctly), then add the two expressions together.

Do you get it now?

 

[5ymmetrica1]

Yes I understand now, I thought that was the case but I was confused by the subtraction statement in your previous post, but it was an error on my part and not in your explanation.

thanks again for all the help curious.

 

[Curious3141]

Yes I understand now, I thought that was the case but I was confused by the subtraction statement in your previous post, but it was an error on my part and not in your explanation.

thanks again for all the help curious.

You're welcome.