MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.2 – Volumes by Slicing; Disks and Washers Copyright © 2006 by Ron.

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Volumes by Slicing: Disks and Washers

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Presentation transcript:

MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.2 – Volumes by Slicing; Disks and Washers Copyright © 2006 by Ron Wallace, all rights reserved.

Reminder: Definition of a Definite Integral Simplified Version where …

Review Process for Area 1. Divide the region into n pieces. 2. Approximate the area of each piece with a rectangle. 3. Add together the areas of the rectangles. 4. Take the limit as n goes to infinity. 5. The result gives a definite integral. This basic approach works with all applications that lead to integrals!

Volumes of Solids Slicing 1. Divide the solid into n pieces (slices). 2. Approximate the volume of each slice. 3. Add together the volumes of the slices. 4. Take the limit as n goes to infinity. 5. The result gives a definite integral.

Volume of a Slice Volume of a cylinder? h r What if the ends are not circles? A What if the ends are not perpendicular to the side? No difference! (note: h is the distance between the ends) A slice is a very short cylinder!

Volume of a Solid A slice is a very short cylinder! axkxk b A(x k ) The hard part? Finding A(x).

Volumes by Slicing: Example A pool is 75’ long and 25’ wide. The depth of the pool gradually increases from 3’ to 10’. How much water does the pool hold? 75’ 25’ 3’ 10’ 0x75 d Another example: See problem 18 in the textbook.

Solids of Revolution A solid obtained by revolving a region around a line. Examples … Sphere: Rotate a semicircle around its diameter. Cone: Rotate a right triangle around one of its legs. Cylinder: Rotate a rectangle around one of its sides. Donut: Rotate a circle around a line that does not intersect the circle. (called a torus)

Solids of Revolution A solid obtained by revolving a region around a line. ab x f(x) Cross-section is perpendicular to the axis of rotation. Disks

Solids of Revolution - Example A solid obtained by revolving a region around a line. Find the volume of a cone of height h and bottom radius r. r h NOTE: Cross-section is perpendicular to the axis of rotation.

Solids of Revolution A solid obtained by revolving a region around a line. When the axis of rotation is NOT a border of the region. Creates a “pipe” and the slice will be a washer. Find the volume of the solid and subtract the volume of the hole. Washers f(x) g(x) xkxk b a NOTE: Cross-section is perpendicular to the axis of rotation.