Answer :
To calculate the volumes using the washer method, we'll find the volumes of revolution for the given functions by revolving the regions around the x-axis. Here is the detailed procedure for each part:
(d) For [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = 2x \)[/tex] from [tex]\( a=0 \)[/tex] to [tex]\( b=2 \)[/tex]:
1. Set up the washer method formula: The volume [tex]\( V \)[/tex] is given by
[tex]\[ V = \pi \int_{a}^{b} [(g(x))^2 - (f(x))^2] \, dx \][/tex]
2. Substitute the given functions and limits:
[tex]\[ V_d = \pi \int_{0}^{2} [(2x)^2 - (x^2)^2] \, dx \][/tex]
3. Simplify the integrand:
[tex]\[ V_d = \pi \int_{0}^{2} [4x^2 - x^4] \, dx \][/tex]
4. Integrate each term separately:
[tex]\[ \pi \left[ \int_{0}^{2} 4x^2 \, dx - \int_{0}^{2} x^4 \, dx \right] \][/tex]
5. Find the antiderivatives:
[tex]\[ \int 4x^2 \, dx = \frac{4x^3}{3}, \quad \int x^4 \, dx = \frac{x^5}{5} \][/tex]
6. Evaluate the definite integrals:
[tex]\[ V_d = \pi \left[ \left. \frac{4x^3}{3} \right|_{0}^{2} - \left. \frac{x^5}{5} \right|_{0}^{2} \right] \][/tex]
7. Substitute the limits:
[tex]\[ V_d = \pi \left[ \frac{4(2)^3}{3} - \frac{(2)^5}{5} \right] = \pi \left[ \frac{32}{3} - \frac{32}{5} \right] \][/tex]
8. Combine fractions to get the final volume:
[tex]\[ V_d = 13.40412865531645 \][/tex]
(b) For [tex]\( f(x) = \tan(x) \)[/tex] and [tex]\( g(x) = \frac{x}{2} \)[/tex] from [tex]\( a=0 \)[/tex] to [tex]\( b=\frac{\pi}{4} \)[/tex]:
1. Set up the washer method formula:
[tex]\[ V_b = \pi \int_{0}^{\frac{\pi}{4}} \left[ \left(\frac{x}{2}\right)^2 - \tan(x)^2 \right] \, dx \][/tex]
2. Simplify the integrand:
[tex]\[ V_b = \pi \int_{0}^{\frac{\pi}{4}} \left[ \frac{x^2}{4} - \tan^2(x) \right] \, dx \][/tex]
3. Evaluate the integral:
This integral is handled numerically and the result is approximately:
[tex]\[ V_b = -0.5473567993669296 \][/tex]
(c) For [tex]\( f(x) = 2\sqrt{x} \)[/tex] and [tex]\( g(x) = 2 \)[/tex] from [tex]\( a=0 \)[/tex] to [tex]\( b=1 \)[/tex]:
1. Set up the washer method formula:
[tex]\[ V_c = \pi \int_{0}^{1} [2^2 - (2\sqrt{x})^2] \, dx \][/tex]
2. Simplify the integrand:
[tex]\[ V_c = \pi \int_{0}^{1} [4 - 4x] \, dx \][/tex]
3. Integrate each term separately:
[tex]\[ \pi \left[ \int_{0}^{1} 4 \, dx - \int_{0}^{1} 4x \, dx \right] \][/tex]
4. Find the antiderivatives:
[tex]\[ \int 4 \, dx = 4x, \quad \int 4x \, dx = 2x^2 \][/tex]
5. Evaluate the definite integrals:
[tex]\[ V_c = \pi \left[ \left. 4x \right|_{0}^{1} - \left. 2x^2 \right|_{0}^{1} \right] \][/tex]
6. Substitute the limits:
[tex]\[ V_c = \pi \left[ 4(1) - 2(1)^2 \right] = \pi \left[ 4 - 2 \right] = 2\pi \][/tex]
7. Simplified result:
[tex]\[ V_c = 6.283185307179585 \][/tex]
Thus, the volumes of revolution for each part are:
- Part (d): [tex]\( 13.40412865531645 \)[/tex]
- Part (b): [tex]\( -0.5473567993669295 \)[/tex]
- Part (c): [tex]\( 6.2831853071795845 \)[/tex]
(d) For [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = 2x \)[/tex] from [tex]\( a=0 \)[/tex] to [tex]\( b=2 \)[/tex]:
1. Set up the washer method formula: The volume [tex]\( V \)[/tex] is given by
[tex]\[ V = \pi \int_{a}^{b} [(g(x))^2 - (f(x))^2] \, dx \][/tex]
2. Substitute the given functions and limits:
[tex]\[ V_d = \pi \int_{0}^{2} [(2x)^2 - (x^2)^2] \, dx \][/tex]
3. Simplify the integrand:
[tex]\[ V_d = \pi \int_{0}^{2} [4x^2 - x^4] \, dx \][/tex]
4. Integrate each term separately:
[tex]\[ \pi \left[ \int_{0}^{2} 4x^2 \, dx - \int_{0}^{2} x^4 \, dx \right] \][/tex]
5. Find the antiderivatives:
[tex]\[ \int 4x^2 \, dx = \frac{4x^3}{3}, \quad \int x^4 \, dx = \frac{x^5}{5} \][/tex]
6. Evaluate the definite integrals:
[tex]\[ V_d = \pi \left[ \left. \frac{4x^3}{3} \right|_{0}^{2} - \left. \frac{x^5}{5} \right|_{0}^{2} \right] \][/tex]
7. Substitute the limits:
[tex]\[ V_d = \pi \left[ \frac{4(2)^3}{3} - \frac{(2)^5}{5} \right] = \pi \left[ \frac{32}{3} - \frac{32}{5} \right] \][/tex]
8. Combine fractions to get the final volume:
[tex]\[ V_d = 13.40412865531645 \][/tex]
(b) For [tex]\( f(x) = \tan(x) \)[/tex] and [tex]\( g(x) = \frac{x}{2} \)[/tex] from [tex]\( a=0 \)[/tex] to [tex]\( b=\frac{\pi}{4} \)[/tex]:
1. Set up the washer method formula:
[tex]\[ V_b = \pi \int_{0}^{\frac{\pi}{4}} \left[ \left(\frac{x}{2}\right)^2 - \tan(x)^2 \right] \, dx \][/tex]
2. Simplify the integrand:
[tex]\[ V_b = \pi \int_{0}^{\frac{\pi}{4}} \left[ \frac{x^2}{4} - \tan^2(x) \right] \, dx \][/tex]
3. Evaluate the integral:
This integral is handled numerically and the result is approximately:
[tex]\[ V_b = -0.5473567993669296 \][/tex]
(c) For [tex]\( f(x) = 2\sqrt{x} \)[/tex] and [tex]\( g(x) = 2 \)[/tex] from [tex]\( a=0 \)[/tex] to [tex]\( b=1 \)[/tex]:
1. Set up the washer method formula:
[tex]\[ V_c = \pi \int_{0}^{1} [2^2 - (2\sqrt{x})^2] \, dx \][/tex]
2. Simplify the integrand:
[tex]\[ V_c = \pi \int_{0}^{1} [4 - 4x] \, dx \][/tex]
3. Integrate each term separately:
[tex]\[ \pi \left[ \int_{0}^{1} 4 \, dx - \int_{0}^{1} 4x \, dx \right] \][/tex]
4. Find the antiderivatives:
[tex]\[ \int 4 \, dx = 4x, \quad \int 4x \, dx = 2x^2 \][/tex]
5. Evaluate the definite integrals:
[tex]\[ V_c = \pi \left[ \left. 4x \right|_{0}^{1} - \left. 2x^2 \right|_{0}^{1} \right] \][/tex]
6. Substitute the limits:
[tex]\[ V_c = \pi \left[ 4(1) - 2(1)^2 \right] = \pi \left[ 4 - 2 \right] = 2\pi \][/tex]
7. Simplified result:
[tex]\[ V_c = 6.283185307179585 \][/tex]
Thus, the volumes of revolution for each part are:
- Part (d): [tex]\( 13.40412865531645 \)[/tex]
- Part (b): [tex]\( -0.5473567993669295 \)[/tex]
- Part (c): [tex]\( 6.2831853071795845 \)[/tex]
