Step by step
The question
Question: find the integral from 0 to 1 of 6e³ˣ.
Two new rules: one for e to a power, one for 1 over x.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The rules
First, e to the x.
Its derivative is eˣ itself: its slope equals its height.
So its integral is eˣ + C too.
With e³ˣ: its derivative is 3 times e³ˣ, by the chain rule.
That 3 must not be there.
So we divide by 3.
Careful!
Do not forget to divide by 3.
2 Integrate
Divide by 3: 6/3 is 2.
No + C: we work out top minus bottom.
3 Top minus bottom
Put in 1, then take away the same with 0.
The 2 in front times both.
e to the 0 is 1.
So the integral is 2 times (e³ − 1).
A calculator gives about 38.171.
4 Check
Let's check with a calculator: it adds up 1000 thin strips.
That gives about 38.171, the same as our answer.
Right!
5 Pick the choice
Our answer is 2(e³ − 1) ≈ 38.171.
That is choice A.
6 Why not the others
B is 114.513.
Wrong: did not divide by 3.
C is 40.171.
Wrong: left out the bottom limit: e⁰ = 1.
D is 3.437.
Wrong: used e¹: the power is 3 × 1 = 3.
Let's look back.
First: the rules.
Then: integrate.
Then: top minus bottom.
Then: check.
Then: pick the choice.
Last: why not the others.