Question

4. [8 points] What is the volume of the solid formed by taking the the area between y = 4- 4x and the x-axis on the interval [0,1], and rotating it about x = 2? Fill in the following blanks to show your work and final answer. Setup but do not solve the definite integral. Volume of slice: Riemann Sum: Sketch of picture: Definite Integral:

Answer 1

y=4-4x²

substuting x=0,we get y intercept =4

and substituting y=0 we get x=+1 and -1

and since it is given that x is in interval [0,1]

so -1 is not a solution of the above equation

so the bounded region is in 1st quadrant

since it is rorating about x=2,we will apply shell method

$V=2\Pi \int_{a}^{b}(h-x)f(x)dx$

here h=2,a=0,b=1

so formula becomes

$V=2\Pi \int_{1}^{1}(2-x)(4-4x^{2})dx$

$V=2\Pi \int_{1}^{1}(-4x+4x^{3}+8-8x^{2})dx$

$V=2\Pi (-\frac{4x^{2}}{2}+\frac{4x^{4}}{4}+8x-\frac{8x^{3}}{3})_{0}^{1}$

$V=2\Pi (-2x^{2}+x^{4}+8x-\frac{8x^{3}}{3})_{0}^{1}$

$V=2\Pi (-2+1+8-\frac{8}{3})$

$V=2\Pi (7-\frac{4}{3})$

$V=2\Pi (\frac{17}{3})$

$V=\frac{34\Pi }{3}$