Question

You are given g(x)=4x^2 + 2x and

f(x) = the integral of g(t) from 0 to x.
How would you find f(6)?

Answer 1

324

Explanation:

Given:

`g(x)=4x^2+2x\\ \\f(x)=\int\limits^x_0 {g(t)} \, dt`

Find:

`f(6)`

First, find f(x):

`f(x)\\ \\=\int\limits^x_0 {g(t)} \, dt\\ \\=\int\limits^x_0 {(4t^2+2t)} \, dt\\ \\=\left(4\cdot (t^3)/(3)+2\cdot (t^2)/(2)\right)\big|\limits^x_0\\ \\=\left((4t^3)/(3)+t^2\right)\big|\limits^x_0\\ \\= \left((4x^3)/(3)+x^2\right)-\left((4\cdot 0^3)/(3)+0^2\right)\\ \\=(4x^3)/(3)+x^2`

Now,

`f(6)\\ \\=(4\cdot 6^3)/(3)+6^2\\ \\=288+36\\ \\=324`