What is the volume

Question

The base of a solid in the region bounded by the parabola x² + y = 4 and the line x + y = 2. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

Question options:

1)
2)
3)
4)

Answer 1

Hi Madison,

First step you need to find intercept between two functions

y = 4 - x^2

y = 2 - x

for finding intercepts we need to solve 4 - x^2 = 2 - x ------> x^2 -x -2 = 0 x =-1 x = 2

V = Π∫ (f^2(x) - g^2(x) ) dx if you graph two functions f(x) = 4 - x^2 and g(x) = 2 - x

V = Π∫ ( ( 4-x^2)^2 - (2-x)^2 )dx

V = pi ∫ (x^4 -8x^2 +16 -4 +4x -x^2) dx

V = pi ∫ ( x^4 -9x^2 +4x +12)dx

V = pi * ( 1/5 X^5 -3x^3 +2x^2 +12x) and bounded of integral is X = 2 , x = -1

V = pi * [32/5 -24 +8 +24 - (-1/5 +3 +2 -12)]

V = pi * (33/5 +15)

V = pi * 108/5 the right answer is choice number 4

I hope it is useful,

Minoo

Answer 2

To get volume we integrate the cross sectional area A(x) with respect to x. So we need 2 things: a function A(x) which tells us the cross sectional area at each value x along our integral, and the limits of integration that we are integrating between. The limits of integration come from where the two curves intersect. Find these points by setting -x² + 4 = -x + 2. We get A(x) by recognizing that the cross sections perpendicular to the x-axis are semicircles. Therefore, at each x value the cross section is a semicircle with a diameter equal to the vertical distance between the parabola and the line. Plugging the limits of integration and A(x) function we just found back into our integral for volume formula will give the answer.