The base of a solid is the circular region bounded by the graph of
x^2 + y^2 =a^2, where a > 0. find the volume of the solid if every cross-section perpendicular to the x-axis is a square?
y = (a^2 - x^2)^.5
1/2 a side of square is y
so area of square at x , A(x) = 4 y^2 = 4 (a^2-x^2)
integrate that dx from x = 0 to x = a and double it for the left half
V = 2 * int(0 to a) (4 a^2 - 4 x^2) dx
= 8 [ a^3 - a^3/3 ] = (16 /3) a^3
check that, I did it fast.
Well, let's crunch some numbers and find our way to the answer, shall we?
First, let's visualize the problem. We have a circular region defined by the equation x^2 + y^2 = a^2, and we want to find the volume of the solid formed by stacking square cross-sections perpendicular to the x-axis.
Now, since the cross-sections are squares, we know that their side length must be equal to the diameter of the circle. The diameter of the circle can be found by solving the equation x^2 + y^2 = a^2 for y = 0. This means we set y to zero and solve for x:
x^2 + 0^2 = a^2
x^2 = a^2
x = a
So, the diameter of the circle is 2a, and that will be the side length of our square cross-sections.
To find the volume of the solid, we need to integrate the area of each square cross-section over the x-axis. The area of a square is just the side length squared. So, the volume V can be calculated by integrating the area function over the range of x = -a to x = a:
V = ∫[-a,a] (2a)^2 dx
V = 4a^2 ∫[-a,a] dx
V = 4a^2 [x]├[-a,a]
V = 4a^2 (a - (-a))
V = 4a^2 (2a)
V = 8a^3
And there you have it! The volume of the solid is 8a^3. I hope this answer didn't square you away too much!
To find the volume of the solid, we need to consider the cross-sections perpendicular to the x-axis. Since every cross-section is a square, we must determine the side length of each square cross-section.
Given that the base is the circular region bounded by the equation x^2 + y^2 = a^2, we can rewrite the equation as y = √(a^2 - x^2).
The side length of each square cross-section can be determined by finding the y-coordinate of the upper half of the circle when x is fixed. This can be done by evaluating y = √(a^2 - x^2) at a fixed x-value.
Since each cross-section is a square, the side length will be twice the y-coordinate, as the circle is symmetric across the x-axis.
Therefore, the side length of each cross-section will be 2√(a^2 - x^2), and the area of each cross-section will be (2√(a^2 - x^2))^2 = 4(a^2 - x^2).
To find the volume of the solid, we'll integrate the cross-sectional areas with respect to x over the entire range of x-values that span the circular region. This range of x-values is from -a to a, as the circle is symmetric around the y-axis.
Therefore, the volume (V) of the solid can be calculated using the following integral:
V = ∫[from -a to a] 4(a^2 - x^2) dx
Now, let's calculate the integral step-by-step:
V = 4 ∫[from -a to a] (a^2 - x^2) dx
V = 4 [ a^2x - (x^3)/3 ] [from -a to a]
V = 4 [ (a^2a - (a^3)/3) - (a^2(-a) - ((-a)^3)/3) ]
V = 4 [ (a^3 - (a^3)/3) - (-a^3 - (-a^3)/3) ]
V = 4 [ (3a^3)/3 + (3a^3)/3 ]
V = 4 [ (6a^3)/3 ]
V = 8a^3
Therefore, the volume of the solid is 8a^3.
To find the volume of the solid, we need to determine the cross-sectional area and integrate it along the x-axis.
First, let's determine the equation of the cross-section perpendicular to the x-axis that forms a square. Since the cross-section is a square, its area is given by the side length squared.
Let's consider a cross-section at a specific x-coordinate, denoted as x. The radius of the circular base at that x-coordinate is given by the equation of the circle:
r(x) = sqrt(a^2 - x^2)
The length of one side of the square cross-section is equal to 2 times the radius, since the square is formed by connecting four points on the circumference of the circle. Therefore, the side length of the square is:
s(x) = 2r(x) = 2sqrt(a^2 - x^2)
The area of the square cross-section is then:
A(x) = s(x)^2 = (2sqrt(a^2 - x^2))^2 = 4(a^2 - x^2)
To find the volume of the solid, we integrate the cross-sectional area function over the interval for which the circle exists ([-a, a]) along the x-axis.
V = ∫[from -a to a] A(x) dx
= ∫[from -a to a] 4(a^2 - x^2) dx
Evaluating the integral:
V = 4 ∫[from -a to a] (a^2 - x^2) dx
= 4 [a^2x - (x^3)/3] evaluated from -a to a
= 4 [a^2(a) - (a^3)/3 - (a^2(-a) - (-a^3)/3)]
= 4 [a^3 - (a^3)/3 + a^3 - (-a^3)/3]
= 4 [2a^3 - 2(a^3)/3]
= 4 (6a^3)/3
= 8a^3
Therefore, the volume of the solid is 8a^3.