Question

What is the arc length of `f(x)= e^(3x) +x^2e^x` on `x in [1,2]`?

Answer 1

The arc length `s` of the curve `f` on `x in[a,b]` is given by

`s=int_a^bsqrt(1+(f'(x))^2)dx`

Here, `f(x)=e^(3x)+x^2e^x` so `f'(x)=3e^(2x)+2xe^x+x^2e^x`. The arc length is then:

`s=int_1^2sqrt(1+(3e^(2x)+2xe^x+x^2e^x)^2)dx`

Put this into an online graphing service like WolframAlpha or into any other calculator to find an approximation for the integral:

`sapproxcolor(blue)97.658636`