Rectangular box of sizes (dimensions) w,l,h wlh Right cylinder of radius r and height h r 2 h

Transcription

1 Volumes: Slicing Method, Method of Disks and Washers -.,.. Volumes of Some Regular Solids: Solid Volume Rectangular bo of sizes (dimensions) w,l,h wlh Right clinder of radius r and height h r h Right cone of radius r and height h r h Sphere of radius r r. Volumes of Solids with Regular (Recognizable) Cross-Sections - slicing method: Let S be a solid. The intersection of S with a plane is a plane region that is called a cross-section of S. Let A be the area of the cross-section of S in a plane P, which is perpendicular to the ais and passing through the point where a b. Then the volume V of S is defined as V lim n n k B the definition of a definite integral, A k Δ,whereΔ b a n a b A d., k a kδ. Similarl, if A is the area of the cross-section of S in a plane P, which is perpendicular to the ais and passing through the point where c d, then the volume V of S is defined as V lim n n k A k Δ c d A d. Eample Show that the volume of a right clinder of radius r and heigh h is r h. Let us place the right clinder horizontall so that the central line of the clinder is on the ais and the beginning of the central line is the origin. For h, the cross-section is a disk whose area is A r and then h r d r h r h. Eample Show that the volume of a sphere of radius r is V r. Let us place the sphere so that the center of the sphere is at the origin. For r, the cross-section

2 is a disk of radius R r and the area of the cross-section is A R r. Then V r A d r r d r r r r r Eample Suppose that the hourglass shape of a swimming pool is defined b curves feet for and the depth of the pool can be described as d. pool. The figure of the pool: Find the volume of the swimming For a given, the cross-section is a retangle with width w and length L are w, L.

3 So, the area of the a cross-section is A wl A d d 5. 5 Eample A church steeple is feet tall with square cross sections. The square at the base has side feet, the square at the top has side inches, and the side varies linearl in between. Find the volume of the steeple. inches foot. Let L be the height of the half triangle. Then.5.5 L L,.5L.5L.5,.5L, L.5 feet.5 Consider the cross-section at where. Let the side of the square be. Then.5,.5 Theareaofthecross-sectionis The volume of the church steeple is A A d.5.5 d 7. 5 feet Eample Let S be a solid that has a circular base of radius. Parallel cross-sections are perpendicular to the base and are equilateral triangles. Find the volume of S. Let. The cross-section is a an equilateral triangle which is perpendicular to the ais and whose base is parallel to the ais and with ends on the circle. See the following picture.

4 The distance between the equilateral triangle and the ais is units. So, the length of the base is b and the heigh h bsin. Then the area of the cross-section is A bh and the volume of the solid is V A d d.. Volumes of Solids Obtained b Rotating a Region H - method of disks and washers: Eample Find the volume of the solid obtained b rotating the region bounded b, and about a. ais; b. ais c. ; d Rotate H about ais Rotate H about ais

5 Rotate H about a. For, A Rotate H about A d / d 5 5/ 5 5/ 9 5 b. For, A A d d c. For, A d. For, A A d d A d d 5. Clindrical Shell Method: The volume of a clinder can be considered as a sum of man thin disks and also can be considered as a sum of man thin clinders. Eample Find the volume of the solid obtained b rotating the region bounded b and 5

6 ais about ais. Find the intersections of the curve and ais.,, For, A rh. A d d 5 Eample Revolve the region bounded b the graphs of and for about a ais; b. Intersections of curves and :,, Region:, a a A rh b d 5 b A rh d Eample Revolve the region bounded b and ais, about.

7 Region:,..... A rh d 7