# Solutions to Problem Set 14

Math 211-03

10-5-2017

[Volumes by cross-sections]

1. The cross-sectional area of a solid in planes perpendicular to the x-axis is

Find the volume of the solid.

Here's the work for the integral:

2. The base of a solid is the region in the x-y plane bounded by and . The cross-sections in planes perpendicular to the x-axis area squares having one edge in the x-y plane. Find the volume of the solid.

Find the intersection points:

A typical square cross-section has sides of length . The volume is

3. The base of a solid is the semicircular region bounded by and the x-axis. The cross-sections in planes perpendicular to the x-axis are equilateral triangles having one edge in the x-y plane. Find the volume of the solid.

A typical triangular cross-section has sides of length . Since the area of an equilateral triangle with sides of length s is , the volume is

4. The base of a solid is the region in the x-y plane bounded by and the x-axis. The cross-sections in planes perpendicular to the x-axis are isosceles right triangles with their hypotenuses lying the in x-y plane. Find the volume of the solid.

Find the x-intercepts:

A typical triangular cross-section has a hypotenuse of length . If the hypotenuse of an isosceles right triangle is h, the legs have length . Therefore, the legs of the cross-section have length . Therefore, the area of a cross-section is

The volume is

5. A solid hemisphere of radius 2 is cut by a plane 1 unit above and parallel to the base.

The top piece is removed. What is the volume of the piece that remains?

If I slice the solid parallel to the base, I get circular cross-sections. Here is a side view:

A typical circular cross-section is shown on edge in the picture. Its radius is

Therefore, its area is

Thus, the volume is

Let us train our minds to desire what the situation demands. - Seneca

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