Question

Find the volume of the solid whose base is bounded by the given function with the indicated cross section taken perpendicular to the x-axis. 1. x2+y2=4, quarter-circles

Answer 1

To find the volume of the solid whose base is bounded by the function x
`^(2)`

= 4 with quarter-circles as cross sections, we use the concept of integration. The volume is obtained by integrating the area of each quarter-circle cross section, which is given by (pi/4) * (4 - x
`^(2)`), and evaluating the integral from -2 to 2. The resulting volume is (4/3) * pi * 8.

Step-by-step explanation:

To find the volume of the solid whose base is bounded by the function x
`^(2)` + y
`^(2)`= 4, with quarter-circles as cross sections taken perpendicular to the x-axis, we can use the concept of integration. First, we need to find the equation of the quarter-circles. From x
`^(2)` + y
`^(2)` = 4, we get y = +sqrt(4 - x
`^(2)`). The volume is obtained by integrating the area of each cross section, which is a quarter-circle. The limits of integration will be -2 to 2, since the base is bounded by x = -2 and x = 2.

The equation of the quarter-circle is y = sqrt(4 - x
`^(2)`), so the area of each quarter-circle is A = (pi/4) * (4 - x
`^(2)`).

Integrating the area formula from -2 to 2 gives:

V = ∫[−2, 2] (pi/4) * (4 - x
`^(2)`) dx

Simplifying the integral and solving:

V = (pi/4)[(8 - (8/3)) - (-8 - (8/3))]

V = (pi/4)[(24/3) + (24/3)] = (4/3)*pi*8

So, the volume of the solid is (4/3)*pi*8.