Use the method of cylindrical shells to find the volume v of the solid obtained by rotating the region bounded by the given curves about the x-axis. x + y = 7, x = 16 − (y − 3)2

Step-by-step explanation:

I put the equations in a plotter, you can see the rotation region below. Now, as we are rotationg about the x-axis we are going to use the equations and the limit points in terms of y. Now, the volume is

where p(y) is the distance from the rotation axis and the differential, a and b are the limit points of the region and h(y) is the height of the region. In this case p(y)= y, a=0, b=7 and h(y) = 16-(y-3)^2-(7-y) (right minus left). So,

.

2 x^2 -3x+p

sum of roots = 3/2

let other root be x

3+ x = 3/2

x= 3/2 - 3 = -3/2

product of roots = p/2

3× -3/2 = p/2

-9 = p

⭐⭐

you can first factor out the n to get n(1+5). we know that 1+5 is 6, so we just have 6n.