This integral represents the volume of a solid.  Find the volume, and describe the solid. `pi int_-1^1 (1-y^2)^2 dy`

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violy | High School Teacher | (Level 1) Associate Educator

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We will apply the foil method here first before integrating it.

Kindly recall that we can rewrite `(1 - y^2)^2` as `(1 - y^2)(1 - y^2)` .
FOIL means first terms, outer terms, inner terms, last terms.
So, we will have:

First terms `= 1 * 1 = 1` .

Outer terms `= 1 * - y^2 = -y^2` .

Inner terms `= -y^2 * 1 = -y^2` .

Last terms` = -y^2 * -y^2 = y^4` .

`(1 - y^2)(1 - y^2) = 1 - y^2 - y^2 + y^4`

Combine like terms.
`(1 - y^2)(1 - y^2) = 1 - y^2 - y^2 + y^4 = 1 - 2y^2 + y^4`

We can now rewrite the problem as` int_-1^1(1 - 2y^2 + y^4)dy` .

We can apply the formula:

`int_a^bu^ndu = (u^{n+1})/(n+1)|_a^b`

So, we will have:
`int_-1^1(1 - 2y^2 + y^4)dy = (y - (2y^3)/(3) + (y^5)/(5))|_-1^1`

Apply the formula: `int_a^bf(x)dx = F(b) - F(a)` .

`(y - (2y^3)/(3) + (y^5)/(5))|_-1^1 = [((1) - (2(1)^3)/(3) + ((1)^5)/(5))] - [((-1) - (2(-1)^3)/(3) + ((-1)^5)/(5))] `
`= [(1 - (2(1))/(3) + (1)/(5)] - [((-1) - (2(-1))/(3) + (-1)/(5))] `
`= [(1) - (2)/(3) + (1)/(5)] - [((-1) - (-2)/(3) + (-1)/(5)] `

Take note that negative times negative is positive.
`[(1) - (2)/(3) + (1)/(5)] - [((-1) - (-2)/(3) + (-1)/(5)]= [(1) - (2)/(3) + (1)/(5)] - [(-1) + (2)/(3) - (1)/(5)] `

Find the lcd, note that identifying the lcd is like finding the least common multiple of
the denominator.
The denominators are `1` , `3` , and `5` .

Multiples of the denominators are:
`1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ..`
`3,6,9,12,15,18,21, ..`
`5,10,15,20,25, ...`
Therefore, the lcd is `15` .

We will make the denominators equal to `15` .
`[(1) - (2)/(3) + (1)/(5)] - [(-1) + (2)/(3) - (1)/(5)] = [(15)/(15) - (2 * 5)/(15) + (1*3)/(15)] - [(-15)/(15) + (2*5)/(15) - (1*3)/(15)] `

Since` 3 * 5 = 15` , we multiply `(2)/(3)` by `5` on top and bottom. Since `5 *3 = 15` , we multiply `(1)/(5)`  by `3` on top and bottom.
`[(15)/(15) - (2 * 5)/(15) + (1*3)/(15)] - [(-15)/(15) + (2*5)/(15) - (1*3)/(15)] = (15 - 10 + 3)/(15) - (-15 + 10 - 3)/(15) = (8)/(15) - (-8)/(15)`

Since, negative times negative is positive we will have:
`(8)/(15) - (-8)/(15) = (8)/(15) + (8)/(15) `
`= (16)/(15).`

Multiply the `pi` outside the integral, the answer will be `(16pi)/(15)` .
That is it! :)

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