# The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and he line x+2y=8. If the cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

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To find the volume of the solid, you integrate the area of each cross-section perpendicular to the x-axis from the starting x-value to the ending x-value.

The equation of the line is (x + 2y = 8), rearranging to (y = \frac{8 - x}{2}). This line intersects the x-axis at (x = 8) and the y-axis at (y = 4).

The region bounded by the x-axis, the y-axis, and the line (x + 2y = 8) is a triangle with base 8 and height 4.

The area of each cross-section perpendicular to the x-axis is a semicircle, and the radius of each semicircle is the value of y. Thus, the area of each cross-section is (\frac{\pi y^2}{2}).

Integrating from (x = 0) to (x = 8), the volume (V) of the solid is given by:

[V = \int_{0}^{8} \frac{\pi y^2}{2} dx]

Substituting (y = \frac{8 - x}{2}) into the integral expression:

[V = \int_{0}^{8} \frac{\pi (\frac{8 - x}{2})^2}{2} dx]

[V = \int_{0}^{8} \frac{\pi (64 - 32x + x^2)}{8} dx]

[V = \frac{\pi}{8} \int_{0}^{8} (64 - 32x + x^2) dx]

[V = \frac{\pi}{8} \left[ 64x - 16x^2 + \frac{x^3}{3} \right]_{0}^{8}]

[V = \frac{\pi}{8} \left[ (64 \times 8 - 16 \times 8^2 + \frac{8^3}{3}) - (0 - 0 + 0) \right]]

[V = \frac{\pi}{8} \left[ (512 - 1024 + \frac{512}{3}) \right]]

[V = \frac{\pi}{8} \left[ \frac{512}{3} \right]]

[V = \frac{\pi}{3} \times 16]

[V = \frac{16\pi}{3}]

So, the volume of the solid is (\frac{16\pi}{3}) cubic units.

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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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