Question

the area labeled b is five times the area labeled a. express b in terms of a.
y
y = e^x
a
x
a
b =

y
y = e^x
b
x
b

Explanation:

Step1: Define Area A via integral

Area \( A \) is the integral of \( y=e^x \) from \( 0 \) to \( a \):
$$A = \int_{0}^{a} e^x dx$$

Step2: Compute Area A

Evaluate the integral:
$$A = e^x \bigg|_{0}^{a} = e^a - e^0 = e^a - 1$$

Step3: Define Area B via integral

Area \( B \) is the integral of \( y=e^x \) from \( 0 \) to \( b \):
$$B = \int_{0}^{b} e^x dx$$

Step4: Compute Area B

Evaluate the integral:
$$B = e^x \bigg|_{0}^{b} = e^b - e^0 = e^b - 1$$

Step5: Set up area relationship

Use \( B = 5A \):
$$e^b - 1 = 5(e^a - 1)$$

Step6: Isolate \( e^b \)

Simplify the right-hand side:
$$e^b = 5e^a - 5 + 1 = 5e^a - 4$$

Step7: Solve for \( b \)

Take natural log of both sides:
$$b = \ln(5e^a - 4)$$

Answer:

\( b = \ln(5e^a - 4) \)