MAT 21B Lecture Notes - Lecture 8: Riemann Sum, Ellipse
Chapter 6. Applications of definite integrals
§6.1: Volume using cross sections
Motivation: Given a solid object, how to calculate its volume?
For simple solids, we may have geometric formulas.
•The volume of a cylinder is
V=A·h
•The volume for a sphere is
V=4
3πr3
•The volume for a cone
V=1
3A·h
•The volume of a cylindrical solid with arbitrary
base is
V=A·h
Question: Given a more complicated solid, how to calculate its volume?
•Slice the solid into sum of smaller
pieces.
V=X
k
Vk
•Each small piece may be approx-
imated by a cylinder.
Vk≈A(xk)∆xk
1
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Document Summary
For simple solids, we may have geometric formulas: the volume of a cylindrical solid with arbitrary base is, the volume of a cylinder is. V = a h: the volume for a sphere is. V = a h: slice the solid into sum of smaller pieces. Vk: each small piece may be approx- imated by a cylinder. P = {x0, x1, , xn} of [a, b], the volume of the solid s can be approximated by a riemann sum. Here, a(x) denotes the area of the cross- section at x. To increase accuracy of approximation, we let the norm ||p|| 0: A(x)dx whenever a(x) is integrable on [a, b]. To apply the formula in the de nition to calculate the volume of a solid, take the following steps: Example: suppose a(x) = a for a positive constant a, then the volume is v =? x2dx. In particular, for a cone, its volume is.
