Chapter 7: Q10E (page 422)
What does the Theorem of Pappus say?
Theorem of Pappussays that
Let R be a plane region that lies entirely on one side of a line L in the plane. If R is rotated about L, then the volume of the resulting solid is the product of the area A of R and the distance d travelled by the centroid of R.
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To determine the volume of the remaining portion of the sphere.
To represent the integral function as an area and to calculate the volume of the solid region (circular disk).
A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced \(1.5\;{\rm{cm}}\) apart. The liver is \(15\;{\rm{cm}}\) long and the cross-sectional areas, in square centimetres, are \({\bf{0}},{\bf{18}},{\bf{58}},{\bf{79}},{\bf{94}},{\bf{106}},{\bf{117}},{\bf{128}},{\rm{ }}{\bf{63}},{\bf{39}}{\rm{ }},\)and \(0\). Use the Midpoint Rule to estimate the volume of the liver.
For what values of\(m\)do the line\(y = mx\).and the curve\(y = \frac{x}{{{x^2} + 1}}\)enclose a region? Find the area of the region.
To show the volume enclosed by the barrel \(V = \frac{1}{3}\pi h\left( {2{R^2} + {r^2} - \frac{2}{5}{d^2}} \right)\).
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