Question

Definite Integral Question?

Answer 1

Suppose \[\int\limits_{1}^{3} g(t)dt = 12\]

Answer 2

Find \[\int\limits_{5}^{15} g(x/5)\]

Answer 3

I know the answer is 16 but i dont know how to get it

Answer 4

same with \[\int\limits_{-1/3}^{1/3} g(2-3t)dt\]

Answer 5

answer is 60 not 16

Answer 6

Let's keep this in our back pocket, we'll come back to it later:
\[\Large\bf\sf \color{#DD4747 }{\int\limits\limits_{1}^{3} g(t)dt \quad=\quad 12}\]
So we need to figure this out:
\[\Large\bf\sf \int\limits\limits_{5}^{15} g\left(\frac{x}{5}\right)dx\]
We'll start by making a substitution:\[\Large\bf\sf u=\frac{x}{5}\]What do you get for your dx?

Answer 7

x^2/10

Answer 8

Hmm no.. maybe I didn't ask the question clearly.
We're just taking the derivative of our u.\[\Large\bf\sf du=\frac{1}{5}dx\qquad\to\qquad 5du=dx\]Does that make sense?

Answer 9

ohh okay yeah

Answer 10

Then we'll also want to change the boundaries of our function:\[\Large\bf\sf x=5:\qquad u=\frac{5}{5}=1\]\[\Large\bf\sf x=15:\qquad u=\frac{15}{5}=3\]

Answer 11

So plugging all of these goodies in gives us:\[\Large\bf\sf 5\color{#DD4747 }{\int\limits\limits_{u=1}^3 g(u)\;du}\]

Answer 12

Hmm it's starting to look familiar, yes? :o

Answer 13

Yeah! thanks you really helped me make sense of this