Find the area a of the figure enclosed by the curve y = e ^ x, y = 2 and x = 0, and the volume of one revolution around the Y axis Using the knowledge of definite integral,

Find the area a of the figure enclosed by the curve y = e ^ x, y = 2 and x = 0, and the volume of one revolution around the Y axis Using the knowledge of definite integral,

If y = e ^ X and y = 2, the intersection of them is (ln2,2)
X = 0 is the Y axis, and the area of the figure surrounded by the three is the definite integral of F (x) = 2 - e ^ X between 0 and LN2
F(x) = ∫(2-e^x)dx = 2x - e^x + C
A = F(ln2) - F(0) = (2ln2 - 2) - (0 - 1) = 2ln2 -1
Y = e ^ x, x = LNY; the radius at Y is LNY, the sectional area is π (LNY) & # 178;, and the integral interval is [1,2]
G(y) = π∫(lny)²dy = π(yln²y -2ylny + 2y) +C
G(2) = π(2ln²2 - 4ln2+ 4) +C
G(1) = 2π + C
V = G(2) - G(1) = 2π(ln²2 - 2ln2 + 1) = 2π(ln2 -1)²