INTEGRATING VOLUMES: DISKS METHOD Made By: Curtis Wei’s Blood, Sweat, and Tears MTED 301

STANDARDS CALIFORNIA COMMON CORE CALCULUS STANDARD 16.0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.

OBJECTIVE STUDENTS WILL BE ABLE TO: Use the DISK METHOD to calculate the VOLUMES of SOLIDS OF REVOLUTION

VOCABULARY CROSS-SECTION DISK METHOD RIEMANN SUM SOLIDS OF REVOLUTION The 2-D shape you get when you cut straight through an 3-D object DISK METHOD The method of calculating volume using integration and disks RIEMANN SUM A method of approximation commonly used to find the area under functions SOLIDS OF REVOLUTION The 3-D shape formed by rotating a function around a line or axis

WARM UP SKETCH the graph of the functions Find the INTERSECTING AREA between the curves using DEFINITE INTEGRAL 𝑓 𝑥 = 𝑥 2 𝑔 𝑥 =4𝑥− 𝑥 2

𝑓 𝑥 = 𝑥 2 𝑔 𝑥 =4𝑥− 𝑥 2 ANSWER Find the intersecting points to make a definite integral (2,4) or x = 2 (0,0) or x = 0

ANSWER 𝑓 𝑥 = 𝑥 2 𝑔 𝑥 =4𝑥− 𝑥 2 THIS AREA Integrate to find the intersecting area THIS AREA 8 3 𝑢𝑛𝑖𝑡 𝑠 2

WAIT A SECOND! WHAT IS THE DISK METHOD? The method and means of calculating the VOLUME using integration with DISKS WE KNOW HOW TO FIND VOLUMES OF SHAPES! Sphere Cone Cylinder Prism 𝑉= 4 3 𝜋 𝑟 3 𝑉=𝜋 𝑟 2 ℎ 𝑉= 1 3 𝜋 𝑟 2 ℎ 𝑉=𝑙×𝑤×ℎ

SO…WHY USE THE DISK METHOD? Where do you think the STANDARD VOLUME FORMULAS come from????? And what happens when you need the VOLUME of weird shapes like THESE SOLIDS OF REVOLUTION * ** This actually uses DONUT METHOD* And this uses WASHER METHOD**

HOW DOES IT WORK? DISK METHOD is the application of the RIEMANN SUM THEOREM on 3-D shapes Instead of deriving AREA with RECTANGLES, the DISK METHOD derives VOLUME with CROSS SECTIONAL AREA of the SOLID OF REVOLUTION Now let’s go through an example! RIEMANN SUM INTERVENTION

𝑓 𝑥 = 𝑥 0≤𝑥≤4 Calculate the volume of the function by rotating it about the x-axis and using the DISK METHOD 2. ROTATE THE FUNCTION TO MAKE A SOLID OF REVOLUTION 1. GRAPH THE FUNCTION

𝑓 𝑥 = 𝑥 0≤𝑥≤4 Calculate the volume of the function by rotating it about the x-axis and using the DISK METHOD 𝐴=𝜋 𝑟 2 =𝜋 𝑓(𝑥) 2 =𝜋𝑥 𝑟=𝑓 𝑥 = 𝑥 3. FIND THE CROSS-SECTIONAL AREA

𝑓 𝑥 = 𝑥 0≤𝑥≤4 Calculate the volume of the function by rotating it about the x-axis and using the DISK METHOD 4. APPLY RIEMANN SUM THEOREM AND SET UP THE INTEGRAL

𝑓 𝑥 = 𝑥 0≤𝑥≤4 Calculate the volume of the function by rotating it about the x-axis and using the DISK METHOD 5. SOLVE THE INTEGRAL = 0 4 𝜋𝑥 𝑑𝑥= 𝜋 𝑥 2 2 0 4 = 𝜋 (4) 2 2 =𝟖𝝅 𝑢𝑛𝑖𝑡 𝑠 3

WE’RE DONE! MORE EXAMPLES TOMORROW! HAVE A GREAT DAY!

RIEMANN SUM????

LET’S REVIEW RIEMANN SUMS! RIEMANN SUMS THEOREM WHAT ARE RIEMANN SUMS? DISCUSS WITH YOUR GROUP If f is integrable on [a,b], then where and

LET’S BREAK IT DOWN 2. Find AREA of each rectangle 3. Take the SUM of the areas (areas under the x-axis represent negative (-) areas) Riemann Sums Theorem: 1. Divide the function into n-number of RECTANGLES 4. Set the n-number of rectangles to approach INFINITY 5. The NET TOTAL is approx. the DEFINITE INTEGRAL of the function The TOTAL SUM of the AREA an INFINITE number of RECTANGLES equals the DEFINITE INTEGRAL of the function

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