Adding and Subtracting Volumes
The region \(R\) is bounded by the curve with equation \(y=x^3+2\), the line \(y=5-2x\).
- \(V_{1}=\pi \int ^1_{0} (x^3+2)^2\, dx\)
- \(V_{1}=\pi \int x^6+4x^4+4\, dx\)
- \(V_{1}=\frac{36}{7}\pi\)
- \(V_{2}=\frac{1}{3}\pi r^2h\)
- \(V_{2}=\frac{1}{3}\pi 3^{2} (\frac{3}{2})\)
- \(V_{2}=\frac{9}{2}\pi\)
- \(V_{\text{total}}=\frac{36}{7}\pi+\frac{9}{2}\pi=\frac{135}{14}\pi\)
The region \(R\) is bounded by curves with equation \(y=\sqrt{ x }\) and \(y=\frac{1}{8x}\)
- Simultaneous Equations to find intercept: \(\frac{1}{8x}=\sqrt{ x }\) = \(0.25\)
- \(V_{1}=\pi \int ^1_{0.25} x^{\frac{1}{2}}\, dx=\frac{15}{32}\pi\)
- \(V_{2}=\pi \int ^1_{\frac{1}{4}} \left( \frac{1}{8x} \right)^2\, dx=\frac{3}{64}\pi\)
- \(V_{\text{total}}=V_{1}-V_{2}=\frac{27}{64}\pi\)
The region \(R\) is bounded by curves \(y=x\) and \(y=\sqrt[3 { x }\)
- Simultaneous Equations: \(\sqrt[3]{ x }=x\) \(x=1\)
- \(V_{1}=\pi \int (\sqrt[3]{ x })^2\, dx=\frac{3}{5}\pi\)
- \(V_{2}=\pi \int x^2\, dx=\frac{1}{3}\pi\)
- \(V_{\text{total}}=\frac{3}{5}\pi-\frac{1}{3}\pi=\frac{4}{15}\pi\)