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Home Applications of the Integral Volumes of Solids of Revolution Disc Method  
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Disc Method
DISC METHOD: For rotations about the axis of the independent variable. Let us first consider the region under a curve. Let R be the region under a curve y = f(x) from x = a to x = b, shown in Figure 6.2.3(a).
x is the independent variable in this case. To keep R in the first quadrant we assume 0 ≤ a < b and 0 ≤ f(x). Rotate R about the xaxis, generating the solid of revolution S shown in Figure 6.2.3(b). This volume is given by the formula below. VOLUME BY DISC METHOD To justify this formula we slice the region R into vertical strips of infinitesimal width Δx. This slices the solid S into discs of infinitesimal thickness Δx. Each disc is almost a cylinder of height Δx whose base is a circle of radius f(x) (Figure 6.2.4). Therefore ΔV = π(f(x))^{2} Δx (compared to Δx). Then by the Infinite Sum Theorem we get the desired formula Figure 6.2.4 Disc Method


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