3). The shaded region is bounded by the y-axis and the graphs of y=1+√x, y=2. Find the volume of the solid obtained by rotating this region around the x-axis.

Question

Answer choices: 7/6pi, 4/3pi, 11/6pi, 5/3pi, 13/6pi, 5/6pi
4). Find the area of the region bounded by y=x^2-6x+7, y=7+6x-x^2

Answer 1

3. I don't see your graph but sketched it.

V= π∫(4 - (1+√x)^2 dx from x = 0 to 1
= π∫ (4 - 1 - 2√x - x) dx from 0 to 1
= π [ 3x - (4/3)x^(3/2) - (1/2)x^2 ] from 0 to 1
= π [ 3 - 4/3 - 1/2 - ( 0 )]
= π[7/6]
= 7π/6 , which is the first choice

4. first find their intersection:
x^2 - 6x + 7 = 7 + 6x - x^2
2x^2 - 12x = 0
x^2 - 6x = 0
x(x-6) = 0
so x = 0 or x = 6

area = ∫7 + 6x - x^2 - (x^2 - 6x + 7) dx from 0 to 6
= ∫( -2x^2 + 12x) dx from 0 to 6
= [(-2/3)x^3 + 6x^2] from 0 to 6
= ((-2/3)(216) + 216 - 0)
= 72