Question

how would you know what the integral of a graph is if you don't have a calculator to graph and you're not good at envisioning what the graph (x^2 + 1) ^3 looks like?
medals given

Answer 1

so basically you want to sketch the function after evaluating integral ?

Answer 2

since the given function is a 6th degree polynomial, integrating it would give u a 7th degree polynomial

Answer 3

i don't necessarily want to sketch it but I want to know what the integral of it is

Answer 4

oh then its easy, just expand it using binomial theorem

Answer 5

and then set it equal to 0? so would the integral be 0 to 1?

Answer 6

why do u want to set it equal to 0 ? whats the exact question ?

Answer 7

I'm not sure why I did that haha.. I'm just trying to find the integral because the question says: a solid is generated when the region in the first quadrant enclosed by the graph y= (x^2 +1)^3, the line x=1, the x-axis, and the y-axis is revolved around the x-axis. Its volume is found by evaluating what integral?

Answer 8

So I know that you would multiply pi times the integral of a to b of the function

Answer 9

oh, okay then you're on right track ! one sec

Answer 10

Volume by revolving around x-axis :
\[\int \limits_0^1 \pi y^2 dx\]

Answer 11

plugin the value of "y" and evaluate

Answer 12

\[\int \limits_0^1 \pi y^2 dx =\int \limits_0^1 \pi ((x^2 +1)^3)^2 dx = \pi \int \limits_0^1 (x^2 +1)^6 dx \]

Answer 13

expand and evaluate

Answer 14

how did you know that the integral of a to b was 0 to 1 though?

Answer 15

read the question :
```
when the region in the first quadrant enclosed by the graph y= (x^2 +1)^3,
the line x=1, the x-axis, and the y-axis is revolved around the x-axis.
```

Answer 16

equation of y axis is : x = 0
and you're given : x = 1