Volume of Solid of Known Cross-Sections Problem: To find the volume of a given solid S. What do we know about the solid? Suppose we are told what the cross-sections perpendicular to some axis are. Figure: Solid whose cross-sections we know We can use the Riemann-sum approach to tackle this problem. This method of finding volume is sometimes called the method of slicing. It turns out that if the axis is labeled as the x-axis, and the cross-sectional area of the solid at point x is A(x) fora apple x apple b, thenthevolumeofthesolids is given by V = Let s see why. Z b a A(x) dx. Math 267 (University of Calgary) Winter 2017 1 / 12
Take a partition P : a = x 0 < x 1 <...<x n = b of [a, b]. Cut the solid into slices using the planes perpendicular to the x-axis at locations x i (i =0, 1, 2,...,n). Figure: Finding volume by slicing On each sub-interval [x i 1, x i ], i =1, 2,...,n, takeasamplepointx i. Multiply the cross-sectional area at the sample points A(x i )andthethickness x i = x i x i 1 of the associated slides to approximate the volumes of the slices that make up the solid: Volume of the i-th slice (or slab) A(x i ) x i. The volume of the solid is approximated by their sum: Volume of the solid nx i=1 A(x i ) x i. Taking limit as the slices get arbitrarily thin, we get Volume of the solid = lim max x i!0 nx i=1 A(x i ) x i = Z b a A(x) dx. Math 267 (University of Calgary) Winter 2017 2 / 12
Finding volumes of solids with known cross-sections To find the volume of a solid with described cross-sections: Example Sketch a typical cross-section. Express the area of the cross-section A(x) asafunctionofx. Integrate A(x) overtheintervala apple x apple b with appropriate endpoints. Suppose that the base of a solid S is the semicircular region R below the graph of y = p 4 x 2 and that the cross-section perpendicular to x is an isosceles right-angled triangle so that the right angle is always made at the vertex on the x-axis. Find the volume of S. Solution: We sketch the base of the solid and the cross-section at x: We express the cross-section area in terms of x: A(x) = 1 2 y 2 = 1 2 4 x 2. We compute the volume by integration V = Z 2 A(x) dx = Z 2 2 2 1 2 4 x 2 dx = 1 4x 2 1 2 3 x 3 2 = 16 3. Math 267 (University of Calgary) Winter 2017 3 / 12
Figure: 3D views of the solid Math 267 (University of Calgary) Winter 2017 4 / 12
Solid of Revolution Suppose we are given a region R and a straight line l on the same plane such that the line does not go through the interior of the region. If we rotate R about the line l, weobtainasolid of revolution. The line l is called the axis of rotation. Figure: 3D views of the solid Depending on whether the axis of rotation coincides with parts of the border of the region, the resulting solid may or may have a hollow central part. The cross-sections perpendicular to the axis of rotation are disks or washers, depending on whether the cross-section hits a hollow central part (thus, washer) or not (thus, disk). Math 267 (University of Calgary) Winter 2017 5 / 12
Volume of a Solid of Revolution Whether disk or washer, we can easily write down the formula for the cross-sectional area A(x) andhence compute the volume of the solid. Figure: Formulas for the cross-sectional area and V = Z b a V = Z b (outer radius) 2 a (radius) 2 dx (cross-sections being disks) (inner radius) 2 dx (cross-sections being washers) This method of finding the volume of a solid of revolution is called the disk method or the disk/washer method. If we visualize the region as composed of infinitely many line segments perpendicular to the axis of rotation, then each of these line segments will produce a disk/washer cross-section when being rotated about the axis. Math 267 (University of Calgary) Winter 2017 6 / 12
Finding the volume of a solid of revolution by the Disk/Washer Method To find the volume of a solid of revolution using the disk/washer method: Example Sketch a typical disk or washer cross-section. For a disk, find its radius. For a washer, find its inner and outer radii. Write down the cross-sectional area function and integrate. Let R be the region bounded below the graph of y = 1 p x 2 +1 between x =0andx =1. Findthevolumeof the solid obtained by rotating R about the x-axis. Solution: We sketch the region, the axis of rotation and the resulting solid of revolution. Figure: Solid of revolution Math 267 (University of Calgary) Winter 2017 7 / 12
We take a typical line segment perpendicular to the axis of rotation, rotate it about the axis to obtain the cross-section there. It is a disk. Figure: The Solid of revolution with cross-sectional disks The radius of the cross-section is r = y = 1 p x 2 +1. The cross-section area is: A(x) = r 2 = We compute the volume by integration V = Z 1 0 A(x) dx = Z 1 0 x 2 +1. x 2 +1 dx = tan 1 (x) 1 0 = tan 1 (1) tan 1 (0) = 2 4. Math 267 (University of Calgary) Winter 2017 8 / 12
In the example above, we rotate a region about a horizontal line. To get a cross-section, we take a line segment perpendicular to the axis of rotation, thus, a vertical line segment. The cross-sections and their areas are functions of x, i.e.,a(x). The disks have thickness x i,andinthelimit,dx. To find volume, we integrate with respect to dx. Imagine what happens if we are to rotate a region about a vertical line. To get a cross-section, we take a line segment perpendicular to the axis of rotation, thus, a horizontal line segment. Figure: Rotation about a vertical axis The cross-sections and their areas are functions of y, i.e.,a(y). The disks have thickness y i,andinthelimit,dy. To find volume, we integrate with respect to dy. Math 267 (University of Calgary) Winter 2017 9 / 12
Example Let R be the bounded region in the first quadrant enclosed by the curves y = x and y = x 2.Findthevolume of the solid obtained by rotating R about the y-axis. Solution: We sketch the region, the axis of rotation and the resulting solid of revolution. We also sketch a horizontal line segment and the resulting disk/washer cross-section. We express everything in terms of y and integrate the cross-sectional area function with respect to dy. The cross-sections are washers at location y (0 apple y apple 1) with thickness dy. The outer radius is the x-coordinate from the curve y = x 2,thus,r outer = p y. The innder radius is the x-coordinate from the curve y = x, thus,r inner = y. The cross-sectional area is A(y) = r 2 outer r 2 inner = y y 2. Z 1 Z 1 Volume V = A(y) dy = y y 2 dy = 6. 0 0 Math 267 (University of Calgary) Winter 2017 10 / 12
Further examples/exercises Find the volume of the following solids. Let S be the solid whose base is the bounded region R enclosed the graphs of y = x 2 and y = p x and whose cross-sections perpendicular to the x-axis are isosceles triangles of height equal to their x-coordinate. Awedgeiscutoutofacircularcylinderofradiusrbytwoplanes.Oneoftheplanesisperpendicularto the axis of the cylinder, cutting out a cross-section that is a disk D. Theotherplanemakesanacute angle with the first plane, meeting it along a diameter of the disk D: Whatisthevolumeofthe wedge? Try slicing the solid in three di erent ways. The solid formed by intersecting two circular cylinders with perpendicular axes and equal diameters. Express the answer in terms of the common radius r Math 267 (University of Calgary) Winter 2017 11 / 12
Consider the following solids of revolution. Let R be the region in the first quadrant bounded below the graph of y = e x between x =0and x =1. FindthevolumeofthesolidobtainedbyrotationR about the x-axis. Let R be the region in the first quadrant bounded below the graph of y =sin(x) betweenx =0and x =. FindthevolumeofthesolidobtainedbyrotationR about the line y = 1; about the line y =2. Find the volume of a ball/sphere of radius r. [Hint: Rotate a semicircular region about its diameter edge.] Derive the formula for the volume of a cone with base radius r and height h. [Hint: Consider the triangular region in the first quadrant with vertices (0, 0), (0, h) and(r, 0). Rotate it about the y-axis.] Consider a disk D and a line l lying on the same plane without intersecting. A donut-shaped solid (called a torus) isobtainedwhenwerotated about l. SupposetheradiusofD is a and the distance from the centre of D to the axis l is b, whereb > a. Whatisthevolumeofthetorus? Math 267 (University of Calgary) Winter 2017 12 / 12