Intergral area calculus

Question

Find the area of the region enclosed by the graphs of 𝑦=𝑒𝑥, 𝑦=𝑒−𝑥, and 𝑦=7.

(Use symbolic notation and fractions where needed.)

Answer 1

Please draw a graph of the reagion.

You can see that it is symmetrical around the y-axis.

So it is sufficient to calculate the area of the right half and then double the result.

The line y=7 intersects the curve y = e^x at x = ln(7).

So the area of the right half is the definite integral from 0 to ln(7)

∫(7 - e^x)dx = 7x - e^x + C

So the area of the right half is

7ln(7) - e^ln(7) - (0 - e⁰) = 7ln(7) - 6

So the area of the whole region is 14ln(7) - 12 ≈ 15.24