Volume of Cross-Sectional Solids

Problem: Lets take the curve y = 2sin(x) and create a solid by constructing semi-circular cross sections perpendicular to the x axis (the XY plane) whose diameters extend from the x-axis to the curve. You can think of an individual cross section as a semicircular fin sticking up out of the XY plane. What would be the volume of this solid?

2sin(x) 2sin(x) dx

This sum can be represented by the following integral: In the diagram above we only have 7 semi-circular cross sections. But the 3D object we are designing has an infinite number of these cross sections, so that the figure will look like the one above The sum of the volumes of an infinite number of semi-circular cross sectional fins with diameter 2sin(x) and thickness dx, on the interval This sum can be represented by the following integral: will give us the volume of the final 3D object.

Recall that the volume of each cross sectional disk was: Why? Recall that the volume of each cross sectional disk was: dx 2sin(x) If we want to sum up an infinite number of these disks We simply integrate this volume over the desired interval:

Steps for Finding Volume of Cross Sections Find an expression for the area of a single generic cross section. Let’s call that expression A(x). 2) Multiply A(x) by dx (the thickness of the cross-sectional disk) to get the volume of the cross sectional disk. 3) Integrate A(x)dx over the interval in question:

Another Example (taken from Problem 1 of the 2000 AP): Let R be the region in the first quadrant enclosed by the graphs of : and the y axis (as shown). The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.

Let’s try to picture what this solid looks like: 1) Here is our region R 2) Now let’s isolate our region R and lay it flat on a 3 dimensional coordinate system 3) Now let’s add square cross sections perpendicular to the X axis R 4) Here is what our final solid looks like

So what would be the volume of this solid? 1) Let’s take one square cross section and find its area: Since our 2 curves intersect at dx