find the exact area

Question

Find the exact area of the surface obtained by rotating the curve about the x-axis.

y= sqrt 1 + e^x,    0 ≤ x ≤ 6

Answer 1

feel free to reach out for a session :).

your formula is the circumference of the circle times the infinitesimal arc length:

2Pi*INT(sqrt(1+dy/dx^2)*y. sqrt(1+dy/dx^2) is for arc length, 2Piy is circle circumference.

dy/dx is (1/2)*e^x/sqrt(1+e^x)

this condenses nicely when squared and add 1 to get

2Pi*INT(1+(1/2)e^x)

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Pi(11+e^6) ~ 1301.97

which compares nicely to the upper bound (in this case) of the truncated cone area of 1328.41