Warm-Up! Find the average value of 𝑓 𝑥 = sin 𝑥 on 0, 𝜋

Applications of Integrals Area and Volume AP Calculus AB

6.1 – Area Between Two Curves

Suppose we are given two functions 𝑦=𝑓 𝑥 and 𝑦=𝑔 𝑥 such that 𝑓 𝑥 ≥𝑔 𝑥 for all 𝑥 in an interval 𝑎, 𝑏 . The graph of 𝑓 𝑥 lies above the graph of 𝑔 𝑥 and the area between the graphs is equal to the integral of 𝑓 𝑥 −𝑔 𝑥 : Area between the graphs= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 − 𝑎 𝑏 𝑔 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑥 −𝑔 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑦 top − 𝑦 bot 𝑑𝑥 The figure illustrates this formula in the case that both graphs lie above the 𝑥-axis.

Example Find the area of the region between the graphs of the functions on the given interval. 𝑓 𝑥 = 𝑥 2 −4𝑥+10 𝑔 𝑥 =4𝑥− 𝑥 2 1≤𝑥≤3

Example Find the area of the region between the graphs of the functions on the given interval. 𝑓 𝑥 = 𝑥 2 −5𝑥−7 𝑔 𝑥 =𝑥−12 −2≤𝑥≤5

Example Find the area of the region between the graphs of the functions. 𝑦= 8 𝑥 2 𝑦=8𝑥 𝑦=𝑥

Homework Text pages 361-363 #s 1-3, 7, 8, 28

6.2 – Setting Up Integrals: Volume, Density, Average Value

6.3 – Volumes of Revolution

A solid of revolution is a solid obtained by rotating a region in the plan about an axis. Suppose that 𝑓 𝑥 ≥0 for 𝑎≤𝑥≤𝑏. The solid obtained by rotating the region under the graph about the 𝑥-axis has a special feature: All vertical cross sections are circles. In fact, the vertical cross section at location 𝑥 is a circle of radius 𝑅=𝑓 𝑥 and thus Area of the vertical cross section=𝜋 𝑅 2 =𝜋 𝑓 𝑥 2 Therefore 𝑉= 𝑎 𝑏 𝜋 𝑓 𝑥 2 𝑑𝑥

Volume of Revolution: Disk Method If 𝑓 𝑥 is continuous and 𝑓 𝑥 ≥0 on 𝑎, 𝑏 , then the solid obtained by rotating the region under the graph about the 𝑥-axis has volume [with 𝑅=𝑓 𝑥 ] 𝑉=𝜋 𝑎 𝑏 𝑅 2 𝑑𝑥 =𝜋 𝑎 𝑏 𝑓 𝑥 2 𝑑𝑥

Example Calculate the volume 𝑉 of the solid obtained by rotating the region under 𝑦= 𝑥 2 about the 𝑥-axis for 0≤𝑥≤2.

Example Calculate the volume 𝑉 of the solid obtained by rotating the region under 𝑦=3𝑥− 𝑥 2 about the 𝑥-axis for 0≤𝑥≤3.

Washer Method Consider the region between two curves 𝑦=𝑓 𝑥 and 𝑦=𝑔 𝑥 , where 𝑓 𝑥 ≥𝑔 𝑥 ≥0 When this region is rotated about the 𝑥-axis, it becomes a washer. The inner and outer radii of this washer are 𝑅 outer =𝑓 𝑥 𝑅 inner =𝑔 𝑥 The volume of the solid of revolution is 𝑉=𝜋 𝑎 𝑏 𝑅 outer 2 − 𝑅 inner 2 𝑑𝑥=𝜋 𝑎 𝑏 𝑓 𝑥 2 − 𝑔 𝑥 2 𝑑𝑥 Keep in mind that the integrand is 𝑓 𝑥 2 − 𝑔 𝑥 2 not 𝑓 𝑥 −𝑔 𝑥 2

Example Find the volume 𝑉 obtained by revolving the region between 𝑦= 𝑥 2 +4 and 𝑦=2 about the 𝑥-axis for 1≤𝑥≤3

Example Find the volume 𝑉 obtained by revolving the region between the graphs of 𝑓 𝑥 = 𝑥 2 +2 and 𝑔 𝑥 =4− 𝑥 2 about the horizontal line 𝑦=−3

Example Find the volume 𝑉 obtained by revolving the region between the graphs of 𝑓 𝑥 =9− 𝑥 2 and 𝑦=12 for 0≤𝑥≤3 about the horizontal line 𝑦=15

Homework Text pages 381-382 #s 6, 8, 39, 40

Warm-Up! Let region 𝑅 be the area under the curve of −𝑥 2 +16 on the interval [0,4]. Determine the area of the solid generated by revolving the region around the line 𝑦=17. (Sketch a picture to help with setting up the integral)

FRQ #2 Page AP6-3 Let 𝑅 be the region in the first quadrant bounded above by 𝑦=4𝑥 and below by 𝑦= 𝑥 3 . Set up, but do not evaluate, an integral expression for each of the following. the area of 𝑅 the volume of the solid obtained by rotating 𝑅 about the 𝑦-axis the volume of the solid obtained by rotating 𝑅 about the line 𝑦=20

FRQ #5 2013 – Non-Calculator Let 𝑓 𝑥 =2 𝑥 2 −6𝑥+4 and 𝑔 𝑥 =4 cos 1 4 𝜋𝑥 . Let 𝑅 be the region bounded by the graphs of 𝑓 and 𝑔, as shown in the figure. Find the area of 𝑅.

FRQ #5 2013 – Non-Calculator Let 𝑓 𝑥 =2 𝑥 2 −6𝑥+4 and 𝑔 𝑥 =4 cos 1 4 𝜋𝑥 . Let 𝑅 be the region bounded by the graphs of 𝑓 and 𝑔, as shown in the figure. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when 𝑅 is rotated about the horizontal line 𝑦=4.

FRQ #5 2013 – Non-Calculator Let 𝑓 𝑥 =2 𝑥 2 −6𝑥+4 and 𝑔 𝑥 =4 cos 1 4 𝜋𝑥 . Let 𝑅 be the region bounded by the graphs of 𝑓 and 𝑔, as shown in the figure. The region 𝑅 is the base of a solid. For this solid, each cross section perpendicular to the 𝑥-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid.

FRQ #3 2011 – Non-Calculator Let 𝑅 be the region in the first quadrant enclosed by the graphs of 𝑓 𝑥 =8 𝑥 3 and 𝑔 𝑥 = sin 𝜋𝑥 , as shown. Write an equation for the line tangent to the graph of 𝑓 at 𝑥= 1 2 .

FRQ #3 2013 – Non-Calculator Let 𝑅 be the region in the first quadrant enclosed by the graphs of 𝑓 𝑥 =8 𝑥 3 and 𝑔 𝑥 = sin 𝜋𝑥 , as shown. Find the area of 𝑅.

FRQ #3 2011 – Non-Calculator Let 𝑅 be the region in the first quadrant enclosed by the graphs of 𝑓 𝑥 =8 𝑥 3 and 𝑔 𝑥 = sin 𝜋𝑥 , as shown. Write, but do not evaluate, an integral expression for the volume of the solid generated when 𝑅 is rotated about the horizontal line 𝑦=1.

FRQ #1 2010 Form B – Calculator In the figure, 𝑅 is the shaded region in the first quadrant bounded by the graph of 𝑦=4 ln 3−𝑥 , the horizontal line 𝑦=6, and the vertical line 𝑥=2. Find the area of 𝑅.

FRQ #1 2010 Form B – Calculator In the figure, 𝑅 is the shaded region in the first quadrant bounded by the graph of 𝑦=4 ln 3−𝑥 , the horizontal line 𝑦=6, and the vertical line 𝑥=2. Find the volume of the solid generated when 𝑅 is revolved about the horizontal line 𝑦=8.

FRQ #1 2010 Form B – Calculator In the figure, 𝑅 is the shaded region in the first quadrant bounded by the graph of 𝑦=4 ln 3−𝑥 , the horizontal line 𝑦=6, and the vertical line 𝑥=2. 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 the solid.