Find the area enclosed between 𝑓(𝑥)=4𝑥 and 𝑔(𝑥)=7cos⁡(𝑥) from 𝑥=−0.1 to 𝑥=0.1.

To find the area enclosed between two curves, we need to integrate the absolute difference of the two functions over the given interval.

The absolute difference between the two functions is |f(x) - g(x)| = |4x - 7cos(x)|.

We integrate this from x = -0.1 to x = 0.1.

∫ from -0.1 to 0.1 |4x - 7cos(x)| dx

This integral can be difficult to solve analytically, so it's often solved numerically using techniques like Riemann sums, trapezoidal rule, Simpson's rule, or numerical integration functions on a calculator or computer.

If you're using a calculator or computer, you would enter the integral as it is. If you're doing it by hand, you would need to split the integral into sections where 4x > 7cos(x) and sections where 4x < 7cos(x), solve each section separately, and add the results together.

This is because the absolute value function |a| is equal to a when a > 0 and -a when a < 0. So you need to find where 4x - 7cos(x) changes from positive to negative or vice versa, and split the integral at those points.

In this case, since the interval from -0.1 to 0.1 is very small, you might find that 4x is always either greater than or less than 7cos(x) over the entire interval, so you wouldn't need to split the integral.

Once you've solved the integral, the result is the area enclosed between the two curves.