Question

Find the volume V of the described solid S. The base of S is the region enclosed by the parabola y = 4 - 3x² and the x-axis. Cross-sections perpendicular to the y-axis are squares.

Answer 1

The volume of the solid S can be found by calculating the integral of y^2 over the range of y-values, which can be determined by solving the equation of the parabola y = 4 - 3x^2. The integral evaluates to (8/3)√(4/3).

Step-by-step explanation:

To find the volume of the solid S, we need to find the area of each cross-section perpendicular to the y-axis and then integrate over the range of y-values. Since the cross-sections are squares, the area of each cross-section is y^2. The range of y-values can be found by setting 4 - 3x^2 = 0 and solving for x, which gives us x = ±√(4/3). So, the volume V of the solid S is given by the integral of y^2 from -√(4/3) to √(4/3):

V = ∫-√(4/3)⁺√(4/3) y^2 dy

To solve this integral, we can use the power rule for integration: ∫ y^n dy = (1/(n+1))y^(n+1). Plugging in the values, we get:

V = (1/3)y^3 |-√(4/3)⁺√(4/3)

V = (1/3)(√(4/3))^3 - (1/3)(-√(4/3))^3

V = (1/3)(4√(4/3)) - (1/3)(-4√(4/3))

V = (8/3)√(4/3)

So, the volume of the solid S is (8/3)√(4/3).