Calculus 5.9: Volume of a Solid by Plane Slicing

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Calculus 5.9: Volume of a Solid by Plane Slicing

Jelly Bean Jar Volume of a solid of uniform cross-sectional area = (area of cross section) (length of solid)

Objective: Given a solid whose cross-sectional area varies along its length, find its volume by slicing it into slabs and performing the appropriate calculus, and show that your answer is reasonable. Technique: Cut the solid into flat slices, formed either by strips in a rotated region or by planes passed through the solid. Get disks, washers, or slabs whose volumes can be found in terms of the solid’s cross section at sample point(s) (x,y) Use geometry to get dV in terms of the sample point(s). Use algebra to get dV in terms of one variable. Use calculus to add up all the dV’s and take the limit (that is, integrate). Check your answer to make sure it’s reasonable

Example 1: Disk Problem The region under the graph of from x=0 to x=8 is rotated about the x-axis to form a solid. Write an integral for the volume of the solid and evaluate it to find the volume. Assume that x and y are in centimeters. Show that your answer is reasonable.

Example 2: Disk Problem The region in Quadrant 1 bounded by the parabola y = 4-x2 is rotated about the y-axis to form a solid paraboloid. Find the volume of the paraboloid if x and y are in inches. Show your answer is reasonable.

HW 5.9A: Q1-Q10 & 1-4

Example 3: Washer Let R be the region bounded by the graphs of y1=6e-0.2x and and by the vertical lines x=1 and x=4. Find the volume of the solid generated when R is rotated about the x-axis. Assume that x and y are in feet. Show that your answer is reasonable.

Vocab Solid of Revolution: A figure generated by rotating a region Finding Volumes by Plane Slices: Slice with planes perpendicular to the axis of rotation

HW 5.9B: 5-11,14

Example 4: Slab A 2in by 2in by 4in wooden block is carved into the shape shown. The graph of is drawn on the back of the block. Then wood is shaved off the front and top faces in such a way that the remaining solid has square cross sections perpendicular to the x- axis. Find the volume of the solid. Show that your answer is reasonable.

HW 5.9C: 17,19,21