Find the areas of the surfaces generated by revolving the...

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like.
$$y=(1 / 2)\left(x^{2}+1\right), \quad 0 \leq x \leq 1 ; \quad y \text { -axis }$$

Key Concepts

Differentiation and Arc Length

Differentiation is crucial for finding the derivative of the function representing the curve. The derivative is then used in computing the arc length element, which in the context of surfaces of revolution, appears under the square root in the integrand. This step accounts for the slant of the curve when calculating the true length of the surface.

Axis of Revolution

The axis about which the curve is revolved determines how the distance from the curve to the axis is computed, directly affecting the formula used. For example, revolving around the y-axis means that the distance from a point on the curve to the axis is typically given by its x-coordinate, which is then incorporated into the integrand to compute the surface area.

Surface Area of Revolution

This concept involves computing the area of a surface formed when a curve is revolved around an axis. The formula generally derives from integrating the product of the circumference of a representative circular slice and the differential arc length. It is a key method in calculus to find areas of three-dimensional shapes generated from two-dimensional curves.

Definite Integration

Definite integration is used to add up infinitely many small pieces that make up the surface area. In problems involving surfaces of revolution, the definite integral sums the contributions from each infinitesimal segment of the curve over the specified interval, resulting in the total surface area.

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