The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
The volume of the solid S where parallel cross sections perpendicular to the base are isosceles triangles with hypotenuse lying along the base.
Use the definition of volume.
2) Definition of volume:
Let S be a solid that lies between and . If the cross sectional area of S in the plane , through and perpendicular to the is , where is continuous function, then the volume of is
Fig (a) represents the graph of a circle that is the base of solid S and has radius
The red line is the hypotenuse of an isosceles right triangle represented in fig (b).
From fig (a), the hypotenuse of the triangle can be found by finding the distance between the top half of the circle and the and multiplying that by .
The equation of circle with radius is
From fig (a),
Now, find the length of the other two sides of an isosceles triangle by Pythagoras theorem.
Since it is an isosceles triangle, length of the other two sides is equal.
Therefore, substitute in the above equation.
Hence, side length
Area of an isosceles triangle
From fig (a), to find volume, integrate the area from
That is, (because function is even)
Applying the Fundamental Theorem of Calculus,
The volume of the solid S where parallel cross sections perpendicular to the base are isosceles triangles with hypotenuse lying along the base is .