he base of a solid in the xy-plane is the circle x^2 + y^2 = 16. Cross sections of the solid perpendicular to the y-axis are semicircles. What is the volume, in cubic units, of the solid?
ayight
To find the volume of the solid, we need to integrate the area of the cross-sections along the y-axis.
The equation of the circle x^2 + y^2 = 16 can be rewritten as y = sqrt(16 - x^2) or y = -sqrt(16 - x^2).
These equations represent the top and bottom halves of the circle, respectively.
Since the cross-sections are semicircles, we only need to consider the top half.
To find the range of y-values for the cross-sections, we need to solve the equation for y in terms of x:
sqrt(16 - x^2) = y
Square both sides to eliminate the square root:
16 - x^2 = y^2
Rearrange the equation to solve for x:
x^2 = 16 - y^2
x = sqrt(16 - y^2)
The range of y-values is from -4 to 4, as given by the equation of the circle.
The area of a semicircle can be calculated using the formula A = (π * r^2) / 2, where r is the radius.
In this case, the radius is given by r = sqrt(16 - y^2).
Thus, the area of each cross-section at a given y value is A = (π * (16 - y^2)) / 2.
To find the volume, we integrate the area over the range of y-values:
V = ∫[from -4 to 4] (π * (16 - y^2)) / 2 dy
Integrating this expression, we get:
V = (π/2) * ∫[from -4 to 4] (16y - y^3/3) dy
Integrating:
V = (π/2) * [(16y^2/2) - (y^4/12)] [from -4 to 4]
V = (π/2) * [(8 * 16/2 - 4^4/12) - (-8 * 16/2 - (-4)^4/12)]
V = (π/2) * [(64 - 64/3) - (-64 + 64/3)]
V = (π/2) * [(128/3) - (32/3)]
V = (π/2) * (96/3)
V = π * 32
Therefore, the volume of the solid is 32π cubic units.
To find the volume of the solid, we need to integrate the area of the cross sections as we move along the y-axis from the bottom of the solid to the top.
The base of the solid is a circle with radius 4, centered at the origin (0, 0). Since the cross sections are semicircles, the radius of each semicircle will be a function of y.
To determine the radius of the cross section at a specific value of y, we need to evaluate the equation of the circle for that value of y. Since the equation of the circle is x^2 + y^2 = 16, we can solve for x^2 by subtracting y^2 from both sides:
x^2 = 16 - y^2
Taking the square root of both sides, we get:
x = √(16 - y^2)
The radius of each semicircular cross section is thus √(16 - y^2).
To find the volume, we integrate the area of the cross sections over the range of y-values. Since the cross-sections are perpendicular to the y-axis, the integration will be with respect to y. The limits of integration determine the range of y-values that correspond to the height of the solid.
Since the solid is formed by the semi-circular cross sections as y varies from the bottom of the solid to the top, the limits of integration will be the y-coordinates of the bottom and top of the semi-circular region, which are -4 and 4 respectively (matching the radius of the base circle).
Therefore, we need to integrate the area of each cross section from y = -4 to y = 4:
V = ∫[from -4 to 4] π⋅(√(16 - y^2))^2 dy
Simplifying:
V = ∫[from -4 to 4] π⋅(16 - y^2) dy
Now we can integrate:
V = π⋅[16y - (y^3)/3] [from -4 to 4]
Evaluating the definite integral gives:
V = π⋅[(16⋅4 - (4^3)/3) - (16⋅(-4) - ((-4)^3)/3)]
V = π⋅[64 - (64/3) - (-64 + (64/3))]
V = π⋅[64 - (64/3) + 64 - (64/3)]
V = π⋅[(192 - 64 - 64)/3]
V = π⋅[64/3]
V = 64π/3
Therefore, the volume of the solid is 64π/3 cubic units.
circles stacked to make hemisphere of radius 4
Volume = (1/2) (4/3) pi r^3
= (2/3) pi (4^3)