To find the exact area of the surface obtained by rotating the curve about the x-axis, we can use the formula for the surface area of a solid of revolution:

A = ∫[a,b] 2πf(x)√[1 + (f'(x))^2] dx

Here, f(x) = 5 - x, and the interval [a,b] is [3,5].

First, we need to find the derivative of f(x), f'(x):

f'(x) = -1

Then, we substitute f(x) and f'(x) into the formula:

A = ∫[3,5] 2π(5 - x)√[1 + (-1)^2] dx
= ∫[3,5] 2π(5 - x)√2 dx

Now, we can compute the integral:

A = 2π√2 ∫[3,5] (5 - x) dx
= 2π√2 [5x - 0.5x^2] evaluated from 3 to 5
= 2π√2 [(5*5 - 0.5*5^2) - (5*3 - 0.5*3^2)]
= 2π√2 [12.5 - 7.5]
= 2π√2 * 5
= 10π√2

So, the exact area of the surface obtained by rotating the curve y = 5 - x, 3 ≤ x ≤ 5 about the x-axis is 10π√2 square units.

Question

To find the exact area of the surface obtained by rotating the curve about the x-axis, we can use the formula for the surface area of a solid of revolution:

A = ∫[a,b] 2πf(x)√[1 + (f'(x))^2] dx

Here, f(x) = 5 - x, and the interval [a,b] is [3,5].

First, we need to find the derivative of f(x), f'(x):

f'(x) = -1

Then, we substitute f(x) and f'(x) into the formula:

A = ∫[3,5] 2π(5 - x)√[1 + (-1)^2] dx
  = ∫[3,5] 2π(5 - x)√2 dx

Now, we can compute the integral:

A = 2π√2 ∫[3,5] (5 - x) dx
  = 2π√2 [5x - 0.5x^2] evaluated from 3 to 5
  = 2π√2 [(5*5 - 0.5*5^2) - (5*3 - 0.5*3^2)]
  = 2π√2 [12.5 - 7.5]
  = 2π√2 * 5
  = 10π√2

So, the exact area of the surface obtained by rotating the curve y = 5 - x, 3 ≤ x ≤ 5 about the x-axis is 10π√2 square units.

Knowee AI · Accepted Answer

To find the exact area of the surface obtained by rotating the curve about the x-axis, we can use the formula for the surface area of a solid of revolution:

A = ∫[a,b] 2πf(x)√[1 + (f'(x))^2] dx

Here, f(x) = 5 - x, and the interval [a,b] is [3,5].

First, we need to find the derivative of f(x), f'(x):

f'(x) = -1

Then, we substitute f(x) and f'(x) into the formula:

A = ∫[3,5] 2π(5 - x)√[1 + (-1)^2] dx
  = ∫[3,5] 2π(5 - x)√2 dx

Now, we can compute the integral:

A = 2π√2 ∫[3,5] (5 - x) dx
  = 2π√2 [5x - 0.5x^2] evaluated from 3 to 5
  = 2π√2 [(5*5 - 0.5*5^2) - (5*3 - 0.5*3^2)]
  = 2π√2 [12.5 - 7.5]
  = 2π√2 * 5
  = 10π√2

So, the exact area of the surface obtained by rotating the curve y = 5 - x, 3 ≤ x ≤ 5 about the x-axis is 10π√2 square units.

Find the exact area of the surface obtained by rotating the curve about the x-axis.y = 5 − x, 3 ≤ x ≤ 5

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