$(15$ points$)$ CONSIDER THE REGION BOUNDED BY $y=e^{-x^2}$ the $x-$axis between $x=0$ and $x=3.$

a$.$ Find the area of the region by evaluating an integral using SERIES. Use $10$ terms and state the error of the approximation.

b$.$ Revolve the region around the $y-$axis and write an integral for the volume of it$.$ Evaluate the volume by hand $($no tech, except to get a decimal approx. to your exact answer after integration$)$ using any appropriate integration technique.

c$.$ Write an integral used to find the arc length of the curve between $x=0$ and $x=2.$ Evaluate the integral using technology $($not by hand$).$ Round the answer to $5$ decimal places. State which program $($cocalc or geogebra$)$ you used.