Two functions are given as f(x)=x^2-3x+5 and g(x)=〖-x〗^2+6?

A

Clearly for there to be a numerical area of the surface between the two functions, they need bound an area i.e. the area needs to be enclosed otherwise it would be infinite. So you have to find where those bounds are.

This means where they intersect. Set them equal to each other and solve.

f(x) = g(x)

x^2 - 3x + 5 = -x^2 + 6

I'm just gonna ignore the thing around the x.

Solve for x here using quadratic formula or whatever, and those will be x-coordinates of the meeting points. This will be our bounds of integration.

Similar to how you find area under ONE function by integrating it, you find area between TWO or more functions by integrating their difference.

Figure out which function is larger (in the interval between the bounds), f(x) or g(x). Then subtract larger - smaller and integrate that with the bounds you find. This will be the area.

Which function is larger is important because that governs the sign. If you subtract smaller - larger, the area will turn out to be negative which is not really an option.

B

Square larger function. Square smaller function.

Subtract square of larger - square of smaller.

Multiply that by pi.

Integrate with given bounds.

This is done because rotating functions around x-axis yields a set of concentric circle slices varying in radius. The cross-sectional area of a single slice is the difference between areas of the two circles which is where the pi comes in.