How can I find distance traveled with a gyroscope and accelerometer?

Basic calculus behind this problem is in the expression

(and similar expressions for displacements in y and z) and basic geometry is the Pythagorean theorem

So, once you have your accelerometer signals passed through a low-pass filter and binned in time with sampling interval dt, you can find the displacement in x as (pardon my C…)

float dx=0.0f;
float vx=0.0f;
for (int i=1; i<n; i++)
 {
   vx+=(acceleration_x[i-1] + acceleration_x[i])/2.0f*dt;
   dx+=vx*dt;
 }

and similarly for dy and dz. Here

float acceleration_x[n];

contains x-acceleration values from start to end of measurement at times 0, dt, 2*dt, 3*dt, … (n-1)*dt.

To find the total displacement, you just do

dl=sqrt(dx*dx + dy*dy + dz*dz);

Gyroscope is not necessary for this, but if you are measuring linear distances, you can use the gyroscope reading to control that rotation of the device was not too large. If rotation was too strong, make the user re-do the measurement.