appliaction of integrals
TRANSCRIPT
AP Calculus Activ-ity
Today’s teacher: Maria Kim
Volume of a Solid using “Cross” Sec-tional Areas
Contents
1. Basic Concepts! 2. Question 3. Finding the equation of A(y) 4. Computing the volume of the
Solid Body5. Answer
Fundamental Concept: Appli-cation of Integrals
Find areas by slicing the region and adding up the volumes of the slices.
1 How do we use definite integrals to find the volume of a solid?
2Important Equations:
Volume =
When the solid by extends from height y=a to y=b
3 We are given a “boundary” for the base of the shape which will be used to find a length.
4 We will use the given length to find the area of a figure generated from the slice .
5 dy or dx is used to represent the thickness.
6 The volumes from the slices will be added together to get the total volume of the figure.
Fundamental Concept: Appli-cation of Integrals
A chocolate cylinder cake is made like figure 2. The base of the solid is the region between the x-axis and root (9-x2). The vertical cross section of the solid perpendicular to the x-axis are special triangles(30,60,90). Compute the volume of the solid.
Figure 1 Figure 2 Figure 3
Looks so deli-cious!
• Base is a semicir-cle
-3 3
• Since the two solids are exactly the same, we are going to compute one solid and double the volume.
1
First, we find a formula for A(x). Note that the area of the triangle is A(y)= bh.
Base
Base: Height Root 3: 1= root():h h=???
Let’s find it~!
2Then, we need to find the area of the triangle and substitute the equation for A(x).
Volume =
A(x)= Area of triangle= Baseheight
=
3Find the volume of the solid.
Answer: 6 root 3 = 12root3 or 20.7846096908…
Two identical solids! Don’t forget!
GREAT JOB!