Question

calculate the area bounded by y=e^x and y=12-e^x

Answer 1

Let y₁=eˣand y₂=12-eˣ.

First we find out where they intersect, y₁=y₂:

eˣ=12-eˣ, 2eˣ=12, eˣ=6, x=ln(6).

y₁=y₂=6.

As x→−∞, y₁→0. The x-intercept of y₂ is eˣ=12, x=ln(12).

The line x=ln(6) divides the area into two parts. On the right side of x=ln(6), y₂ is the only curve contributing to the area, while on the left side y₁ alone contributes to the area:

PART 1=∫[ln(6),ln(12)](12-eˣ)dx=(12x-eˣ)[ln(6),ln(12)]=

12ln(12)-12-(12ln(6)-6)=12ln(12/6)-6=12ln(2)-6.

PART 2=∫[-∞,ln(6)]eˣdx=(eˣ)[-∞,ln(6)]=6.

So the total area is the sum of the parts:=12ln(2)=8.3178 approx.