Finding the Area Between Curves - Calculus

Question

Let R be the area of the region between x = e^{y − 3} and  for 0.5 < x < 1.5. What is R?

A)

B)

C)

D)

Answer 1

Translate x = e^(y-3) to y − 3 = ln |x| or y = ln |x| + 3 and translate x = √y to y = x².

Use a graphing calculator to show the curves for y = ln |x| + 3 and y = x². The

plot will show that the curve for y = x² is to the right of the curve for y = ln |x| + 3

on the x-y plane.

Going back to x = e^(y-3) and x = √y, one would then construct the integral sought

as ∫(√y − e^(y-3))dy with the limits of integration computed as y = (0.5)² or 0.25 and

y = (1.5)² or 2.25.

The final expression would then be ∫_{(from y=0.25 to y=2.25)}(√y − e^(y-3))dy.

This would be (B) among the choices above.