面积 e^x,e^{-4x},ln(4)

解答

面积

e^{x}, e^{- 4 x}, ln (4)

0.08314 \dots

求解步骤

要找到相交点，解

f_{i} (x) = f_{j} (x)

e^{x} = e^{- 4 x} : x = 0

e^{x} = ln (4) : x = ln (2 ln (2))

e^{- 4 x} = ln (4) : x = - \frac{ln ( 2 ln ( 2 ) )}{4}

= \int_{- \frac{l n ( 2 l n ( 2 ) )}{4}}^{0} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{0}^{l n (2 l n (2))} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- \frac{l n ( 2 l n ( 2 ) )}{4}}^{0} e^{- 4 x} - ln (4) d x + \int_{0}^{l n (2 l n (2))} ∣ e^{x} - ln (4) ∣ d x

\int_{- \frac{l n ( 2 l n ( 2 ) )}{4}}^{0} e^{- 4 x} - ln (4) d x = 0.01662 \dots

\int_{0}^{l n (2 l n (2))} ∣ e^{x} - ln (4) ∣ d x = 0.06651 \dots

= 0.01662 \dots + 0.06651 \dots

化简

= 0.08314 \dots

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