Answer
a) sketch the curves $y=1-x^2$ and $ y=x^6-x+1$ with a computer. These points are near (0,1) and (0.75, 0.43).
b) $A \approx 0.12 $
c) V $ \approx 0.543 $
d) V $ \approx 0.308 $
Work Step by Step
Step 1
A) First, sketch the curves $y=1-x^2$ and $ y=x^6-x+1$ with a computer.
Move the cursor to the points of intersection of these curves to estimate.
These points are near (0,1) and (0.75, 0.43).
Step 2
B) The area of shaded region R
A = $ \int _{0}^{0.75} \left[ \left( 1-x^{2} \right) - \left( x^{6}-x+1 \right) \right] dx $
=$ \int _{0}^{0.75} \left[ 1-x^{2}-x^{6}+x-1 \right] dx $
=$ \left[ \frac{x^{2}}{2}-\frac{x^{3}}{3}-\frac{x^{7}}{7} \right] _{0}^{0.75} $
=$ \left[ \frac{ \left( 0.75 \right) ^{2}}{2}-\frac{ \left( 0.75 \right) ^{3}}{3}-\frac{ \left( 0.75 \right) ^{7}}{7} \right] $
A $ \approx 0.121556$ or $A \approx 0.12 $
Step 3
C) use the slicing method.
The outer radius of the washer is (1-x2)
The inner radius of the washer is (x6-x+1)
The area of the washer
A(x) = $ \pi \left[ \left( 1-x^{2} \right) ^{2}- \left( x^{6}-x+1 \right) ^{2} \right] $
= $ \pi \left[ 1+x^{4}-2x^{2}- \left( x^{12}-2x^{7}+2x^{6}-2x+x^{2}+1 \right) \right] $
= $ \pi \left[ 1+x^{4}-2x^{2}-x^{12}+2x^{7}-2x^{6}+2x-x^{2}-1 \right] $
= $ \pi \left[ x^{4}-3x^{2}-x^{12}+2x^{7}-2x^{6}+2x \right] $
Step 4
The volume of the solid
V= $ \pi \int _{0}^{0.75} \left[ x^{4}-3x^{2}-x^{12}+2x^{7}-2x^{6}+2x \right] dx $
= $ \pi \left[ \frac{1}{5}x^{5}-x^{3}-\frac{1}{13}x^{13}+\frac{2}{8}x^{8}-\frac{2}{7}x^{7}+\frac{2}{2}x^{2} \right] _{0}^{0.75} $
= $ \pi \left[ \frac{1}{5} \left( 0.75 \right) ^{5}- \left( 0.75 \right) ^{3}-\frac{1}{13} \left( 0.75 \right) ^{13}+\frac{2}{8} \left( 0.75 \right) ^{8}-\frac{2}{7} \left( 0.75 \right) ^{7}+ \left( 0.75 \right) ^{2} \right] $
V $ \approx 0.173 \pi $
or V $ \approx 0.543 $
Step 5
D) Use the cylindrical method:
Radius of the shell is x
Circumference of the shell is 2$ \pi x $
The height of the shell $(1- x^{2} ) - ( x^{6}-x+1 ) =1-x^{2}-x^{6}+x-1 $
$ = x-x^2-x^6$
Step 6
Volume V = $ \int _{0}^{0.75} \left( 2 \pi x \right) \left( x-x^{2}-x^{6} \right) dx $
= 2$ \pi \int _{0}^{0.75} \left( x^{2}-x^{3}-x^{7} \right) dx $
= 2$ \pi \left[ \frac{1}{3}x^{3}-\frac{1}{4}x^{4}-\frac{1}{8}x^{8} \right] _{0}^{0.75} $
=2$ \pi \left[ \frac{1}{3} \left( 0.75 \right) ^{3}-\frac{1}{4} \left( 0.75 \right) ^{4}-\frac{1}{8} \left( 0.75 \right) ^{8} \right] $
V $ \approx 0.308 $
