volume of solid whose base is a circle with radius a, and cross sections of the solid cut perpendicular to the x-axis are squares
To find the volume of a solid whose base is a circle with radius a, and whose cross sections are squares cut perpendicular to the x-axis, you can use calculus and integration.
Step 1: Visualize the solid
First, let's visualize the solid. It has a circular base with a radius a, and when you cut perpendicular to the x-axis, the shape of the cross section is a square. As you move along the x-axis, the squares will change in size but maintain their shape.
Step 2: Determine the equation for the shape of the cross section
Since the cross sections are squares, we need to find the side length of each square at a given x-coordinate. Let's call this side length s.
Step 3: Express the side length of the square in terms of x
To express the side length of the square in terms of x, we need to find a relationship between x and s. Since the base of the solid is a circle with radius a, we can use the equation of a circle to relate x and s. The equation of a circle with radius a is x^2 + y^2 = a^2.
Since the cross section is square, the side length of the square is equal to the diameter of the circle.
The diameter of the circle can be expressed as 2y, where y is the height of the square. Therefore, we have the equation 2y = s.
To solve for y, we can rearrange the equation of a circle: x^2 + y^2 = a^2. Solving for y gives us y = sqrt(a^2 - x^2).
Step 4: Express the volume of each square cross section
The volume of each square cross section can be calculated as V = s^2, where s is the side length of the square.
Using the equation 2y = s, we can substitute y = sqrt(a^2 - x^2) into the equation to find s.
So, s = 2*sqrt(a^2 - x^2).
Step 5: Set up the integral for the volume
To find the total volume of the solid, we need to integrate the volume of each square cross section along the x-axis.
The limits of integration will be from -a to a, since the solid is symmetric with respect to the y-axis.
Therefore, the integral for the volume of the solid is:
V = ∫[-a to a] (s^2) dx
= ∫[-a to a] (2*sqrt(a^2 - x^2))^2 dx
= 4∫[-a to a] (a^2 - x^2) dx
= 4[a^2x - (x^3)/3] evaluated from -a to a.
Step 6: Evaluate the definite integral
Evaluating the definite integral gives us:
V = 4[a^2a - (a^3)/3] - 4[a^2(-a) - ((-a)^3)/3]
= 4[3a^3/3 + a^3/3]
= 4[4a^3/3]
= (16/3) * a^3.
Therefore, the volume of the solid is (16/3) * a^3.