the region enclosed by the curves y=2x and y=sqrt (x)+1 from x=0 to x=1 is rotated around the line x=-1 to make a solid. what is the volume

Answer 1

Since we are rotating around the line x=-1, it's best to use the shell method.

V = ∫_a^b 2π•r(x)h(x) dx

V = 2π∫₀¹(x+1)(√(x)+1-2x) dx

V = 2π∫₀¹(x√x + x - 2x² + √x + 1 - 2x) dx

V = 2π∫₀¹(x√x + √x - x - 2x² - 1) dx

V = 9π/5 cubic units

Notes:

-> The rotation around the line x = -1 means our radius is r(x) = x-(-1) = x+1

-> As f(x)=√(x)+1 is the upper function and g(x)=2x is the lower function, the height of the cylinder is h(x) = f(x)-g(x) = √(x)+1-2x

Hope this helped!