Calculus: Early Transcendentals 8th Edition

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 $\approx 0.308$
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 $\approx 0.308$