Area bounded by curves ∣ y ∣ ≥ e − ∣ x ∣ − 2 1 and 2 ∣ x ∣ + ∣ y ∣ + 2 ∣ x ∣ − ∣ y ∣ ≤ 2

Question

Area bounded by curves ∣ y ∣ ≥ e − ∣ x ∣ − 2 1 ​ and 2 ∣ x ∣ + ∣ y ∣ ​ + ​ 2 ∣ x ∣ − ∣ y ∣ ​ ​ ≤ 2

TestHub · Accepted Answer

The second inequality describes the square of side 4. The first excludes a thin central region. Its four equal parts end where the exponential reaches one half.

$max (∣ x ∣, ∣ y ∣) \leq 2, e^{- x} = 1/2 \Rightarrow x = ln 2. E = 4 \int_{0}^{l n 2} (e^{- x} - 1/2) d x = 2 - 2 ln 2.$ $A = 16 - E = 14 + 2 ln 2 = 14 + ln 4.$

Mathematics - Area Under Curves Question with Solution | TestHub

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