{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chap6_Sec2

# Chap6_Sec2 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

This preview shows pages 1–18. Sign up to view the full content.

APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6.2 Volumes APPLICATIONS OF INTEGRATION In this section, we will learn about: Using integration to find out the volume of a solid.
In trying to find the volume of a solid, we face the same type of problem as in finding areas. VOLUMES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
We have an intuitive idea of what volume means. However, we must make this idea precise by using calculus to give an exact definition of volume. VOLUMES
We start with a simple type of solid called a cylinder or, more precisely, a right cylinder. VOLUMES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
As illustrated, a cylinder is bounded by a plane region B 1 , called the base, and a congruent region B 2 in a parallel plane. The cylinder consists of all points on line segments perpendicular to the base and join B 1 to B 2 . CYLINDERS
If the area of the base is A and the height of the cylinder (the distance from B 1 to B 2 ) is h , then the volume V of the cylinder is defined as: V = Ah CYLINDERS

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
In particular, if the base is a circle with radius r , then the cylinder is a circular cylinder with volume V = πr 2 h. CYLINDERS
If the base is a rectangle with length l and width w , then the cylinder is a rectangular box (also called a rectangular parallelepiped) with volume V = l wh. RECTANGULAR PARALLELEPIPEDS

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
For a solid S that isn’t a cylinder, we first ‘cut’ S into pieces and approximate each piece by a cylinder. We estimate the volume of S by adding the volumes of the cylinders. We arrive at the exact volume of S through a limiting process in which the number of pieces becomes large. IRREGULAR SOLIDS
We start by intersecting S with a plane and obtaining a plane region that is called a cross-section of S . IRREGULAR SOLIDS

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Let A ( x ) be the area of the cross-section of S in a plane P x perpendicular to the x -axis and passing through the point x , where a x b . Think of slicing S with a knife through x and computing the area of this slice. IRREGULAR SOLIDS
The cross-sectional area A ( x ) will vary as x increases from a to b . IRREGULAR SOLIDS

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
We divide S into n ‘slabs’ of equal width ∆ x using the planes P x 1 , P x 2 , . . . to slice the solid. Think of slicing a loaf of bread. IRREGULAR SOLIDS
If we choose sample points x i * in [ x i - 1 , x i ], we can approximate the i th slab S i (the part of S that lies between the planes and ) by a cylinder with base area A ( x i * ) and ‘height’ ∆ x . 1 i x P - i x P IRREGULAR SOLIDS

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The volume of this cylinder is A ( x i * ). So, an approximation to our intuitive conception of the volume of the i th slab S i is: ( ) ( *) i V S A xi x IRREGULAR SOLIDS
Adding the volumes of these slabs, we get an approximation to the total volume (that is, what we think of intuitively as the volume): This approximation appears to become better and better as n → ∞.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}