volume integration

Question

What is the approximate volume of the solid created when the region under the curve  on the interval  is rotated around the line ?

Answer 1

We can use the shell method:

V = ∫_a^b 2π⋅r(x)h(x) dx

--> Bounds: [a,b] = [-π/2,π/2]

--> Radius: r(x) = x-(-3) = x+3 (which is 3 units away from the y-axis)

--> Height: h(x) = f(x) = cos(x) + 1

V = ∫[-π/2,π/2] 2π(x+3)(cos(x)+1) dx ≈ 96.92 cubic units

You can also consult Dayv O.'s method if you don't want to use shell method.

Hope this helped!

Answer 2

how about volume=V=12π+6π² which is about 96 cubic units

because area of object is 2+π (cos area -π/2,π/2+box 1 by π)

and x coordinate for centroid of area is 0 (symmetry)

pappus theorem V=2π*area*radius of rotation for centroid

A=2+π

r=3