e^x-4 [-1,3]

Question

I am trying to find the area between the x axis and f(x) using the definite integral 

Answer 1

∫₋₁³ |e^x−4| dx is the area between the function and the x-axis.

With f(x) = e^x−4, the function being integrated, g(x)=|e^x−4|, is equal to f(x) when f(x)≥0, and g(x) = −f(x) = 4−e^x when f(x)≤0. So we need to take the extra pre-calculus step of determining the values of x that make f(x) positive or negative.

Solving e^x−4 = 0 results in x = ln 4. This is the value of x where f(x)=0. Then a "sign-chart" can be used just like in other calculus skills to determine that f(x)≤0 on the interval [-1, ln 4] and f(x)≥0 on the interval [ln 4, 3]. Using the additivity of integration on intervals, we can get rid of those nasty absolute value brackets by splitting apart the definite integral:

∫₋₁³ |e^x−4| dx = ∫₋₁^{ln 4} (−f(x)) dx + ∫_{ln 4}³ f(x) dx

=∫₋₁^{ln 4} (4−e^x) dx + ∫_{ln 4}³ (e^x−4) dx

= (4x-e^x)¦₋₁^{ln 4} + (e^x−4x)¦_{ln 4}³

=(4 ln 4 − 4) − (−4 − 1/e) + (e³ − 12) − (4 − 4 ln 4)

= e³ + 1/e + 8 ln 4 − 16 = 15.544...

Wolfram Alpha gives: http://www.wolframalpha.com/input/?i=Area+between+e^x-4+and+y%3D0+on+[-1%2C3]

The original question was well-written because it used the word between. If the question had been "find the area under f(x)" then we'd need a definition for the area under a function when it dips below the x-axis. Functions can have a signed area of less than zero along these regions, but signed areas are a mathematical convention. Actual areas in the real world can never be negative, so the question "find the area under f(x)" always must assume that f(x)≥0. If f(x) is negative with a poorly-written question that uses the word "under" then an otherwise rational calculus class can deteriorate into a discussion about whether Mauna Loa is taller than Mount Everest, and calculus teachers hate when that happens.

Answer 2

∫_-1³ (e^x - 4) dx = e^x - 4x |_-1³ = (e³ - 12) - (e^-1 + 4) = e³ - 1/e - 16

Answer 3

Hopefully I understand the equation correctly as e^x-4 and not e^x-4. But in either case let's do this!
First we need to know the rule by which e^x is integrated. The rule is as follows:
∫ e^x dx = e^x
In cases were we have e^nx where n is some number, then we'd need to use u substitution. For our purposes, the integral shown will suffice. Since we have a set interval we will add it into our intergral as a lower and upper limit. The whole integral for this equation will look like this:
∫_-1³ e^x-4 dx. When integrating, our e^x remains the same and the 4 changes to 4x. So now we have:

e^x-4x ¦_-1³. We now plug in the limits into the integrated equation:
e³-4(3); e^-1-4(-1).
e³-12; (1/e)+4.

From here we sutract the lower limit equation from the upper:

e³-12-(1/e)-4 = e³-(1/e)-16 = (e⁴-16e-1)/e.

Of course since you're probably allowed a calculator you can find the exact value of this answer which is ≈3.71766

I hope this helped!