I am new to integration, so please do not mark this question as "not enough research done"
Here is the question (please open image in new tab to see it clearly) -
I am getting stuck with the integration steps. Could someone give me hints on how to proceed for each of the given sections?
A
Finding the area of an area divided by two functions has three steps. Finding the right and left edges of the shapes, find the area between the upper line of the shape and the x-axis, and finding the area too much in the previous problem.
For the left and right edges of the shape, what you want is an intersection of the functions, so simply put them equal and solve for $x$. Two of the solutions are going to be limits of integration
For the too large area, simply integrate the function on top in the interval found in the last step.
Now find the area below the shape, this is done by integrating the smaller function on the same interval.
When you have the two areas, simply subtract them to obtain the wanted area.
B
This is very similar to A, but we're going to use the following formula instead of integrating directly. $$ \pi\int_a^b f(t)^2\,\mathrm dt $$ What you must do to solve the problem, is first using the formula to find the area of the shape plus something extra in the middle, and then find the area of that extra part using the same formula. Simply subtract them for the final result.
C
If you subtract 2 from both equations, it's simply a revolution around the x-axis and can be solve the same way as problem B.
D
If I understand this part correctly, the wanted shape is a long version of the shape with 2 as the height, that would simply be given by multiplying the area of the base of the shape with 2, however if you really want an integral, you could do it like this: $$\int_0^2R\,\mathrm dx$$ where $R$ is the area from problem A.
E
For the arc length, simply use the known formula below: $$ \int_a^b\sqrt{1+[f'(t)]^2}\,\mathrm dt $$ Where the function is the $g$ function from the problem, and the interval is the one you found in problem A.