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Find the area of the surface generated by revolving the given curve about the y-axis. x = 2√(2-y), -1 ≤ y ≤ 0.

User David Leal
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Final answer:

To find the area of the surface generated by revolving the curve about the y-axis, use the formula for the surface area of revolution and solve the integral.

Step-by-step explanation:

To find the area of the surface generated by revolving the curve about the y-axis, we can use the formula for the surface area of revolution which is given by:

A = 2π∫[a,b] x(y)√(1+(dy/dx)²) dy

In this case, the curve is defined by x = 2√(2-y) and the limits of integration are -1 ≤ y ≤ 0. We first need to express x in terms of y and find dy/dx:

x = 2√(2-y)

Squaring both sides, we get: x² = 4(2-y)

Expanding, we have: x² = 8 - 4y

Rearranging, we get: y = 2 - x²/4

Now, we can find dy/dx:

dy/dx = d(2 - x²/4) / dx = -x/2

Substituting these expressions into the formula, we have:

A = 2π∫[-1,0] x(y)√(1+(-x/2)²) dy = 2π∫[-1,0] 2√(2-y)√(1+(x/2)²) dy

Now, we can solve this integral to find the area of the surface.

User Florianbaethge
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