# How do you find the volume of the region bounded by the graph of #y = x^2+1# for x is [1,2] rotated around the x axis?

See below.

The region is in blue. The rotation is shown by the red circular arrow.

A representative slice has been taken perpendicular to the axis or rotation to use the method of discs. The slice has black borders.

The volume of a representative slice is

In general, the radius

The thickness is a differential. It is the thin side of the slice. IN this case

The values of

The volume we seek is

# = pi int_1^2 (x^4+2x^2+1) dx#

# = 178/15 pi#

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To find the volume of the region bounded by the graph of (y = x^2 + 1) for (x) in ([1,2]) rotated around the x-axis, you use the method of cylindrical shells. The formula for the volume of a solid obtained by rotating a region bounded by a curve around the x-axis is:

[V = \int_{a}^{b} 2\pi x \cdot f(x) , dx]

Where: (a) and (b) are the limits of integration (in this case, 1 and 2). (f(x)) is the function representing the curve ((x^2 + 1) in this case).

So, substituting the given values, the volume can be calculated as follows:

[V = \int_{1}^{2} 2\pi x \cdot (x^2 + 1) , dx]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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