Question

Find the area bounded by y=tanx, x+y=2, and above the x-axis on the interval [0,2].

A) 0.919 B) 0.923 C) 1.013 D) 1.077 E) 1.494

All answers are rounded by three numbers after the decimal.

The answer is D) 1.077.

This is a multiple choice question from my review sheet. I'm trying to understand on how to approach the problem as I can't seem to be able to find the area between these two functions on my graphing calculator (we're allowed to use a graphing calculator for this problem). Any help on this is appreciated.

Thanks!

Answer 1

we would first need to determine of the 2 functions cross between the interval

Answer 2

if there is no crossing, then its simply taking the integral of the differences

Answer 3

I did that, the best I got is 0.923 for that area

Answer 4

I found the intersect point, I believe it is x=0.85353011

Answer 5

they appear to cross once, and also to cross a vertical asymptote

Answer 6

what would be the significance of the vertical asymptote? Can I use that information to be able to calculate another area?

Answer 7

the vertical asymptote requires us to use an improper integral, which is simply taking the limit of the integral as the interval approaches the asymptote

Answer 8

\[\int_{0}^{a}(2-x)-tan(x)dx\\
~~~~~~~~+\lim_{t\to pi/2}\int_{t}^{a}tan(x)-(2-x)dx\\~~~~~~~~~~~~~~~~~~~~+\lim_{t\to pi/2}\int_{t}^{2}(2-x)-tan(x)dx\]

Answer 9

a to t on that middle one

Answer 10

Ahh, we have not learned improper integral in class... would this be the only way to approach a question like this?

Answer 11

it would be the "proper" approach, yes

Answer 12

i think i misread your problem

Answer 13

it's only bound from [0,2] so i think the area under the x-axis is not necessary?