Calculus Area Enclosed by Exponential Functions. Please help!?

Question

Sketch the region enclosed by y=e^5x, y=e^6x, and x=1. Decide whether to integrate with respect to x or y. Then find the area of the region.

I tried treating this like a normal area-between-two curves problem.. But the graph seems to be out of my scope and I don't know how to do the antiderivatives.. Please help and if possible provide a step-by-step solution!

Answer 1

Setting the expressions equal gives us (0, 1) as the only point of intersection, so we're integrating between the lines x=0 and x=1. Thus we have the double integral Int_0^1[ Int_(e^5x)^(e^6x) [ dy ] dx ].

Thus, we have Int_0^1 [ e^(6x) - e^(5x) ] dx = ((1/6)e^6 - (1/5)e^5) - ((1/6)e^0 - (1/5)e^0) = (1/6)e^6 - (1/5)e^5.

Answer 2

The graphs intersect at x = 0, so we want Integ(0,1){[e^(6x) - e^(5x)]dx}

= e^(6x)/6 - e^(5x)/5 = e^6/6 - e^5/5.