7. y = e^x, y = e^(-x), x = ln 2

7. y = e^x, y = e^(-x), x = ln 2

7. y = e^x, y = e^(-x), x = ln 2

Answer

  1. Explanation:
    • Step 1: Determine the area between the curves formula
      • The area (A) between two curves (y = f(x)) and (y = g(x)) from (x=a) to (x = b) is given by (A=\int_{a}^{b}|f(x)-g(x)|dx). Here, for (x\in[0,\ln 2]), (e^{x}\geq e^{-x}), so (A=\int_{0}^{\ln 2}(e^{x}-e^{-x})dx).
    • Step 2: Integrate term - by - term
      • We know that (\int e^{x}dx=e^{x}+C) and (\int e^{-x}dx=-e^{-x}+C).
      • Then (\int_{0}^{\ln 2}(e^{x}-e^{-x})dx=\left[e^{x}+e^{-x}\right]_{0}^{\ln 2}).
    • Step 3: Evaluate the definite integral
      • First, substitute (x = \ln 2) into (e^{x}+e^{-x}): (e^{\ln 2}+e^{-\ln 2}=2 + \frac{1}{2}).
      • Then substitute (x = 0) into (e^{x}+e^{-x}): (e^{0}+e^{0}=1 + 1=2).
      • Now, (\left[e^{x}+e^{-x}\right]_{0}^{\ln 2}=(2+\frac{1}{2})-2=\frac{1}{2}).
  2. Answer: (\frac{1}{2})