Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region.

Question

y=e^1x, y=e^4x, x=1

Answer 1

wrt x:

∫[0,1] e^(4x)-e^x dx
= 1/4 e^(4x) - e^x [0,1]
= (1/4 e^4 - e) - (1/4 - 1)
= 1/4 (3 + e^4 - 4e)

wrt y:

∫[1,e] lny - 1/4 lny dy + ∫[e,e^4] 1- 1/4 lny dy
= (3/4) + (1/4 (e$4 - 4e))
= 1/4 (3 + e^4 - 4e)