calculus problem?

Question

find the volume of the solid found by rotating the region inside the 1st quadrant enclosed by y=x^2 and y=3x about the x-axis

Answer 1

First find the points of intersection of the two curves.

y = x² = 3x
x² - 3x = 0
x(x - 3) = 0
x = 0, 3

So the limits of integration are from 0 to 3.

Use the washer method.

Volume = ∫π(R² - r²)dx = ∫π[(3x)² - (x²)²]dx

= π∫(9x² - x^4)dx

= π(3x³ - x^5/5) | [Evaluated from x = 0 to 3]

= π(81 - 243/5) - 0

= 162π/5 ≈ 101.7876

Answer 2

f (x) = x^3 - 12x + a million . . . the 1st spinoff set to 0 unearths turning or table sure factors f ' (x) = 3x^2 - 12 3x^2 - 12 = 0 3 * (x + 2) * (x - 2) = 0 x = 2 ... x = - 2 . . . the 2nd spinoff evaluated at x = 2 and -2 determines if those factors are min, max, or neither. f ' ' (x) = 6x f ' ' (2) = 6*2 = 12 <== valuable fee shows x=2 is a community minimum f ' ' (-2) = 6*(-2) = -12 <== damaging fee shows x=-2 is a community optimal a.) x = - 2 is a optimal, and x=2 is a minimum ... so x = - infinity to -2 is increasing x = -2 to +2 is reducing x = +2 to + infinity is increasing b.) f (-2) = (-2)^3 - 12*(-2) + a million = 17 f (2) = (2)^3 - 12*(2) + a million = - 15 c.) . . . the 2nd spinoff set to 0 unearths inflection factors, or the place concavity adjustments 6x = 0 x = 0 <=== inflection factor x = - 2 is a optimal, so could be concave down concavity adjustments on the inflection factor(s) ... so x = - infinity to 0 is concave down x = 0 to + infinity is concave up

Answer 3

Find where they intersect that is where we are going to integrate from and to

so you get

(pi) x intigral from 0 to 3 of (3x)^2-(x^2)^2 dx

you get 101.788

Answer 4

Intersection is at x = 3
Limits are between 0 and 3
V = ∫ π(3x)²dx - ∫ π x^(4) dx
V = ∫ 9π.x² dx - ∫ π.x(^4) dx
V = 3π.x³ - π.x^(5) / 5
Inserting limits:-
V = 81π - 243.π / 5
V = 405 π/ 5 - 243.π / 5
V = 162.π / 5
V = 101.8 units³