Chapter Review

Area A between two graphs:

• \(A=\int_{a}^{b}[f(x)-g(x)]~{\it dx}\), where \(f(x)\geq g(x)\) for all \(x\) in the interval \([a,b]\) (p. 405)
• \(A=\int_{c}^{d}[f(y)-g(y)]~{\it dy}\), where \(f(y)\geq g(y)\) for all \(y\) in the interval \([c,d]\) (p. 409)

Volume \(V\) of a solid of revolution: The disk method:

• \(V=\pi \int_{a}^{b}[f(x)]^{2}~{\it dx}\), where the region is revolved about the \(x\)-axis (p. 414)
• \(V=\pi \int_{c}^{d}[g(y)]^{2}~{\it dy}\), where the region is revolved about the \(y\)-axis (p. 416)

Volume \(V\) of a solid of revolution: The washer method:

• \(V=\pi \int_{a}^{b}\{ [ f(x)] ^{2}-[ g(x) ] ^{2}\} ~{\it dx}\), where the region is revolved about the \(x\)-axis (p. 419)
• \(V=\pi \int_{c}^{d}\{ [ f(y)] ^{2}-[ g(y)] ^{2}\} ~{\it dy},\) where the region is revolved about the \(y\)-axis (p. 420)

• Cylindrical shell: The solid region between two concentric cylinders (p. 424)

Volume \(V\) of a solid of revolution: The shell method:

• \(V=2\pi \int_{a}^{b}xf(x)~{\it dx},\) where the region is revolved about the \(y\)-axis (p. 425)
• \(V=2\pi \int_{c}^{d}yg(y)~{\it dy},\) where the region is revolved about the \(x\)-axis (p. 429)

Volume \(V\) of a solid: The slicing method:

• \(V=\int_{a}^{b}A(x)~{\it dx},\) where the area \(A=A(x)\) of the cross section of a solid is known and is continuous on \([a, b]\) (p. 433)

Arc length formulas:

• \(s=\displaystyle \int_{a}^{b}\sqrt{1+[ f' (x) ] ^{2}}~{\it dx}\) (p. 440)
• \(s=\displaystyle \int_{c}^{d}\sqrt{~1+[ g' ( y) ] ^{2}}~{\it dy}\) (p. 441)

• Work is the energy transferred to or from an object by a force acting on the object. (p. 444)

Work formulas:

• \(F\) is a continuously varying force that moves an object from \(a\) to \(b\): \(W=\int_{a}^{b}F(x)~{\it dx}\) (p. 445)
• \(F\) is a spring force; then \(F(x)=-kx\) (Hooke's Law) (p. 446)
• \(F\) is the force required to pump a liquid; \(F(x)=\rho gV(x)\) (p. 448)
• \(F\) is the attraction between two bodies; \(F(x) =G\dfrac{m_{1} m_{2}}{x^{2}}\) (p. 450)

• Pressure \(P\) is the force \(F\) exerted per unit area \(A\): \(P=\dfrac{F}{A}\) (p. 453)

Hydrostatic force:

• \(F=\displaystyle \int_{c}^{d}\rho g( H-y) [ f(y)-g(y)] ~{\it dy}\) (p. 454)

• The center of mass of an object or system of objects is the point that acts as if all the mass is concentrated at that point. (p. 458)
• The moments of a system represent the tendency of the system to rotate about the center of mass. (p. 458)
• A lamina is a thin, flat sheet of material. (p. 460)
• If a lamina has constant density, it is homogeneous and its center of mass is located at the centroid. (p. 460)

The centroid of a homogeneous lamina:

• The lamina of area \(A\) enclosed by the graph of \(f\), the \(x\)-axis, and the lines \(x=a\) and \(x=b\): \[ \bar{x}=\dfrac{1}{A}\displaystyle \int_{a}^{b}[ xf(x) ] ~{\it dx} \quad \hbox{and}\quad \bar{y}=\dfrac{1}{2A}\displaystyle \int_{a}^{b}[f(x)] ^{2}\,{\it dx}, \] where \(A=\int_{a}^{b} f(x)\,{\it dx}\) (p. 461)

Properties of laminas:

• Symmetry principle: If a homogeneous lamina is symmetric about a line \(L\) or a point \(P\), then the centroid of the lamina lies on \(L\) or at a point \(P\). (p. 463)

The Pappus Theorem for Volume

• Let \(R\) be a plane region and let \(V\) be the volume of the solid of revolution obtained by revolving the area \(A\) of \(R\) about a line that does not intersect \(R\). Then the volume \(V\) of the solid of revolution is \(V=2\pi A\,d,\) where \(d\) is the distance from the centroid of \(R\) to the line. (p. 464)