a. around the line y= -1 using the Washer Method

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Finding the intersection points-
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x^2=2-x

x^2+x-2=0

(x+2)(x-1)=0 \space \rightarrow \space x = -2 , x =1

y= x^2 is the bottom function â€śrâ€ť

y= 2-x is the top function â€śRâ€ť

Use equation: \pi \int (R - axis \space of \space rotation)^2 - (r - axis \space of \space rotation)^2

\pi \int_{-2}^1 (2-x+1)^2 - (x^2+1)^2 dx

simplifies to

\pi \int_ {-2}^1 (3-x)^2 - (x^2+1)^2 dx

then square bothâ€¦

\pi \int_ {-2}^1 (x^2-6x+9) - (x^4+2x^2+1) dx

then combine like termsâ€¦

\pi \int_ {-2}^1 (-x^4-x^2-6x+8) dx

Integrateâ€¦

\Pi [(-1/5x^5-1/3x^3-3x^2+8x)] between [1, -2]

\pi [( -1/5(1)^5 - 1/3(1)^3 - 3(1)^2 + 8 (1)) - ( -1/5(-2)^5 - 1/3(-2)^3 - 3(-2)^2 + 8 (-2))]

= \pi [(4.466666667)+(18.93333333)]

= 73.5132681 units^2

b. around the line x= -2 , using the Shell Method Equation: 2\pi \int r * h * d-something

r= x-axis of rotation h = ( top - bottom)

Finding the intersection points-

x^2=2-x

x^2+x-2=0

(x+2)(x-1) x = -2 , x =1

r= (x+2) h= (2-x-x^2) d-something = dx

2\pi\int_{-2}^1 (x+2) (2-x-x^2) dx

foilâ€¦

2\pi\int_{-2}^1 (2x - x^2 - x^3 + 4 - 2x - 2x^2) dx

combine like termsâ€¦

2\pi\int_{-2}^1 ( -x^3 - 3x^2 +4) dx

Integrate

2\pi[ -1/4x^4 - x^3 +4x ] from_{-2}^1

2\pi [( -1/4(1)^4 - (1)^3 +4(1)) - (-1/4(-2)^4 - (-2)^3 +4(-2))

= 2\pi [ (2.75) - (-4) ]

= 2\pi(6.75)

=13.5 \pi

= 42.41150082 units^2

c. around the line y-axis using the Shell Method.

2\pi\int_{-2}^1 r * h * d-something

r= x-axis of rotation h = ( top - bottom)

2\pi\int_{-2}^0 (x+0) (2-x-x^2) dx

foil

2\pi\int_{-2}^0 (2x-x^2-x^3) dx

integrate

2\pi (x^2 - 1/3x^3 - 1/4x^4) between [1, -2]

2\pi[ (0^2 - 1/3 0^3 - 1/4 0^4) - ( -2 ^2 - 1/3 (-2)^3 - 1/4 (-2)^4) ]

2\pi [ 0 - (8/3)]

= absolute value of -16.75516082 units^2