7 3 Volumes by Slicing Little Rock Central

7. 3: Volumes by Slicing Little Rock Central High School, Little Rock, Arkansas Photo by Vickie Kelly, 2001 Greg Kelly, Hanford High School, Richland, Washington

The curve y = x 2 from 0 to 2 is pictured below A solid is constructed such that the cross sections perpendicular to the x-axis are squares. Find the volume of the solid.

The curve y = x 2 from 0 to 2 is pictured below A solid is constructed such that the cross sections perpendicular to the x-axis are squares. Find the volume of the solid. The area of each square is s 2 and each side s is equal to y. Therefore, the area of each square is equal to And each square has a thickness of… dx

The curve y = x 2 from 0 to 2 is pictured below A solid is constructed such that the cross sections perpendicular to the x-axis are squares. The volume of one square is. Adding all of the volumes together, we get the integral:

Method of Slicing (p 439): 1 Sketch the solid and a typical cross section. 2 Find a formula for A(x). (When you multiply A(x) by dx, you will have third length necessary to find the volume) 3 Find the limits of integration. 4 Integrate A(x) to find volume.

Let R be the region in the first quadrant under the curve Find the volume of the solid whose base is the region R and whose cross-sections perpendicular to the x -axis are squares.

Let R be the region in the first quadrant under the curve Adding all of the squares together gives us…

Let R be the region in the first quadrant under the curve Find the volume of the solid whose base is the region R and whose cross-sections perpendicular to the x -axis are isosceles right triangles with hypotenuse lying within the region R. R R

Let R be the region in the first quadrant under the curve We know that the area of such a triangle is We also know by the Pythagorean Theorem that… R

Let R be the region in the first quadrant under the curve Adding all of the triangles together gives us… And since we know that

Find the volume of a sphere of radius r We find the volume by slicing the sphere. If we put the sphere against the xy plane with the center at the origin, we get. . . r Photo by Vickie Kelly, 2001 r Greg Kelly, Hanford High School, Richland, Washington

Find the volume of a sphere of radius r How do we find a formula for each slice of the sphere? What is the shape of each slice of the sphere? Each slice of the sphere is a disk. r r Photo by Vickie Kelly, 2001 r Greg Kelly, Hanford High School, Richland, Washington

Find the volume of a sphere of radius r How do we find a formula for each slice of the sphere? What is the shape of each slice of the sphere? Each slice of the sphere is a disk. r r Which we remember from geometry is the volume of a sphere. r p