Let f(x)= abs of negative abs of x + pi/2 close abs and g(x)= abs of cosx.

the wording might be confusing so here is another description of the equation (the "vertical line" that I refer to is the line you draw as you draw the absolute value sign...for example "line x line" means that x is enclosed by two vertical lines...abs of x....

So here is the description of f(x):

line, negative sign, line, x, line, plus sign, pi/2, line
a)Find the area of f(x) and g(x).

b)Find the volum of solid obtained by spinning the shape from part a) around the x- axis. when x= 3pi/2?

all those words!

f(x) = |x + π/2|
g(x) = |cos x|

Not sure what you mean in part (a). f(x) and g(x) intersect in a single point: (-π/2,0)

See the graphs at

http://www.wolframalpha.com/input/?i=plot+y%3D|x%2B%CF%80%2F2|%2C+y%3D|cos+x|

Maybe I misread your text.

Steve, I plotted the function the way it was originally stated and it showed a giant W along with the |cosx|

http://www.wolframalpha.com/input/?i=plot+y+%3D+%7C+-+%7Cx%7C+%2B+%CF%80%2F2+%7C+%2C+y+%3D+%7Ccos%28x%29%7C

The "solid" still needs a domain, and not well defined

yeah, i misread f(x)

To find the area of f(x) and g(x), we will first compute the definite integral for each function.

a) Area of f(x):
To find the area of f(x), we need to compute the integral of f(x) from its lower bound to its upper bound. In this case, we need to find the integral of f(x) from negative infinity to infinity. However, given the description of f(x), it is clear that f(x) is symmetric about the x-axis, so we can simplify the computation by just finding the integral from 0 to infinity and then doubling it.

To compute the integral of f(x), we will break it down into its components:
1. abs(-abs(x) + pi/2): This can be split into two cases:
a) When -abs(x) + pi/2 >= 0, abs(-abs(x) + pi/2) simplifies to -abs(x) + pi/2.
b) When -abs(x) + pi/2 < 0, abs(-abs(x) + pi/2) simplifies to abs(x - pi/2).
2. abs(cos(x)): This is the absolute value of the cosine function.

Given that the integral is symmetric, we can focus on integrating one side (i.e., from 0 to infinity) and double the result.

b) Volume of solid obtained by spinning the shape:
To find the volume of the solid obtained by spinning the shape formed by f(x) and g(x) around the x-axis, we need to use the method of cylindrical shells. This method involves integrating the product of the circumference of each cylindrical shell and its height over the desired interval.

In this case, we want to find the volume when x = 3pi/2. To do this, we need to construct the solid formed by f(x) and g(x) from negative infinity to 3pi/2, and then use the cylindrical shell method to calculate the volume within that interval.

Note that the cylindrical shells' height is given by the difference between f(x) and g(x) at each x-value within the interval.

To calculate both the area and volume, you can use mathematical software or a graphing calculator that has integral and volume calculation capabilities.