Question

the given curve is rotated about the y-axis. find the area of the resulting surface. y = 1 4 x2 − 1 2 ln(x), 2 ≤ x ≤ 4

Answer 1

To find the area of the surface formed by rotating the given curve about the y-axis, use the formula for the surface area of a solid of revolution.

Step-by-step explanation:

To find the area of the surface formed by rotating the given curve about the y-axis, we can use the formula for the surface area of a solid of revolution. The formula is:

A = 2π∫[a,b] x(y) * √(1 + (dy/dx)^2) dy

First, we need to express the curve in terms of y. Rearranging the equation y = 1/4x^2 - 1/2ln(x) gives us:

x = √(4(y + 1/8ln(4(y + 1/8))))

Next, we can find the derivative dy/dx and substitute it into the formula. Finally, evaluate the integral from the lower limit 2 to the upper limit 4 to get the area of the resulting surface.

Answer 2

The total area of the regions between the curves is 264.33 square units

Calculating the total area of the regions between the curves

From the question, we have the following parameters that can be used in our computation:

y = 14x² - 12ln(x)

Also, we have the interval to be

2 ≤ x ≤ 4

So, the area of the regions between the curves is

`\text{Area} = \int\limits^4_2 {14x^2 - 12\ln(x)} \, dx`

Integrate

`\text{Area} = [(14)/(3)x^3 - (12)/(x)]|\limits^4_2`

Expand using the limits

So, we have

`\text{Area} = [(14)/(3) * 4^3 - (12)/(4)] - [(14)/(3) * 2^3 - (12)/(2)]`

Evaluate

Area = 264.33

Hence, the total area of the regions between the curves is 264.33 square units