Calculating Volume of a Solid
Calculating volume of a solid Quick Rewind: how to calculate area 1. slice the region to smaller pieces (narrow, nearly rectangle) 2. calculate the area of rectangle 3. take the limit (more and more slices) Recipe for calculating volume (motivated by area) 1. slice the solid into smaler pieces (thin cut) 2. calculate the volume of each \thin cut" 3. take the limit (more and more slices) Step 2: volume ≈ area × thickness, because it is so thin Suppose that x-axis is perpendicular to the direction of cutting, and the solid is between x = a and x = b. Z b Volume= A(x)dx, where A(x) is the area of \thin cut"at x a (\thin cut"is called a cross section, a two-dimensional region) n Z b X A(x)dx = lim A(yi ) · ∆x, (Riemann sum), where ∆x = (b − a)=n, x0 = a, n!1 a i=1 x1 = x0 + ∆x, ··· , xk+1 = xk + ∆x, ··· , xn = b, and xi−1 ≤ yi ≤ xi . Slicing method of volume Example 1: (classical ones) Z h (a) circular cylinder with radius=r, height=h: A(x)dx, A(x) = π[R(x)]2, R(x) = r 0 Z h rx (b) circular cone with radius=r, height=h: A(x)dx, A(x) = π[R(x)]2, R(x) = 0 h Z r p (c) sphere with radius r: A(x)dx, A(x) = π[R(x)]2, R(x) = r 2 − x2. −r Example 2: Find the volume of the solid obtained by rotating the region bounded by y = x − x2, y = 0, x = 0 and x = 1 about x-axis.
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