Let R represent the area in Quadrant IV bounded by f(x)=x^3-4x and g(x)=0.

Question

Find the area of R.

Find the volume of the solid generated by revolving R around the x-axis.

Find the average value of f(x) over the interval [-3, -2].

Answer 1

1) Find the roots of x³-4x.

 

x³-4x=0 ⇔x(x²-4)=0 ⇔ x=0; x=2; x=-2. Since we are looking at quadrant IV, we need x<0 and y<0. Thus we are interested in x=-2 and x=0 roots.

 

Area or R is given by the integral:

 

_-2∫⁰((x³-4x)-0)dx=_-2∫⁰(x³-4x)dx=[¼x⁴-2x²]|_-2⁰=0-[¼*(-2)⁴-2(-2)²]=4

 

Volume of the solid of revolution of R around x-axis is best found by the following integral:

 

V=_-2∫⁰π[f(x)-g(x)]²dx=π_-2∫⁰(x⁶-8x⁴+16x²)dx=π*[x⁷/7-8/5*x⁵+16/3*x³]|_-2⁰=

=-π[(-2)⁷/7-8*(-2)⁵/5+16/3*(-2)³]=π[108/3+128/7-256/5]=324π/105.

 

Average value of f(x) on the interval [-3,-2] is given by the following formula:

 

1/(-2-(-3))*_-3∫^-2(x³-4x)dx=_-3∫^-2(x³-4x)dx=[x⁴/4-2x²]|_-3^-2=[(-2)⁴/4-2*(-2)²]-[(-3)⁴/4-2*(-3)²]=

=-4-9/4=-25/4.