# The base of a solid is the region in the first quadrant bounded by the line x+2y=4 and the coordinate axes, what is the volume of the solid if every cross section perpendicular to the x-axis is a semicircle?

drawing refers

the cross section area of the solid at point x, given it is a semi circle, on the x axis is

So the volume of the solid is:

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The base of the solid is a triangle with vertices at (0,0), (4,0), and (0,2). Since each cross-section perpendicular to the x-axis is a semicircle, the radius of each semicircle is determined by the y-coordinate at that point.

The equation of the line x + 2y = 4 can be rewritten as y = (4 - x) / 2.

The y-coordinate ranges from 0 to 2 as x ranges from 0 to 4. So, the radius of the semicircle varies from 0 to 2.

The area of a semicircle with radius r is A = (1/2) * π * r^2.

Using the fact that the radius varies with y, we integrate the area of each semicircle over the range of y (from 0 to 2) to find the volume of the solid.

The integral to find the volume V is given by:

[ V = \int_{0}^{2} \frac{1}{2} \pi \left( \frac{4 - x}{2} \right)^2 , dx ]

Simplifying and integrating gives:

[ V = \frac{1}{8} \pi \int_{0}^{4} (4 - x)^2 , dx ]

After integrating, the volume V of the solid can be calculated.

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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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