Question

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y=√49−x^2​, −4≤x≤4

Answer 1

To find the area of the surface generated by revolving the curve y=√49−x^2 about the x-axis, we can use the formula for the surface area of revolution.

Step-by-step explanation:

The definite integral that represents the area of the surface generated by revolving the curve y=√(49-x^2), on the interval −4≤x≤4 about the x-axis can be calculated using the surface area formula for a function revolved around the x-axis:

A = 2π ∫_b^a y ∞ (1+ (dy/dx)^2) dx

First, compute the derivative of y with respect to x:

√y/√x = (-x)/√(49-x^2)

Then, plug it into the formula:

A = 2π ∫_4^-4 √(49-x^2) ∞(1 + ((-x)^2/(49-x^2))) dx

After simplifying the integrand and evaluating the integral, we find the surface area of the solid formed.

Keep in mind that due to the symmetrical nature of the curve with respect to the y-axis, the integral can also be simplified by evaluating it from 0 to 4 and then doubling the result.

Answer 2

The surface area generated by revolving the curve y=√(49-x^2) from x=-4 to x=4 around the x-axis is determined by setting up and calculating a definite integral involving the circumference of the surface and its derivative with respect to x.

Step-by-step explanation:

The definite integral that represents the area of the surface generated by revolving the curve y=√(49-x^2), on the interval −4≤x≤4 about the x-axis can be calculated using the surface area formula for a function revolved around the x-axis:

A = 2π ∫_b^a y ∞ (1+ (dy/dx)^2) dx

First, compute the derivative of y with respect to x:

√y/√x = (-x)/√(49-x^2)

Then, plug it into the formula:

A = 2π ∫_4^-4 √(49-x^2) ∞(1 + ((-x)^2/(49-x^2))) dx

After simplifying the integrand and evaluating the integral, we find the surface area of the solid formed.

Keep in mind that due to the symmetrical nature of the curve with respect to the y-axis, the integral can also be simplified by evaluating it from 0 to 4 and then doubling the result.