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1.22. To graph f (x) = 3 sin (4 x) − 5, f (x) = 3 sin (4 x) − 5, the graph of y = sin (x) y = sin (x) needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function f f will have a period of π / 2 π / 2 and an amplitude of 3. 1.23.
- 6.8 Exponential Growth and Decay
Learning Objectives. 6.8.1 Use the exponential growth model...
- 3.7 Derivatives of Inverse Functions
This formula may also be used to extend the power rule to...
- 5.7 Integrals Resulting in Inverse Trigonometric Functions
In this section we focus on integrals that result in inverse...
- 6.9 Calculus of The Hyperbolic Functions
Learning Objectives. 6.9.1 Apply the formulas for...
- 4.5 Derivatives and The Shape of a Graph
Using the First Derivative Test. Consider a function f f...
- 2.2 The Limit of a Function
Estimate lim x → 1 1 x − 1 x − 1 lim x → 1 1 x − 1 x − 1...
- 4.7 Applied Optimization Problems
Maximizing the Area of a Garden. A rectangular garden is to...
- 6.4 Arc Length of a Curve and Surface Area
Arc Length of the Curve x = g(y). We have just seen how to...
- 6.8 Exponential Growth and Decay
16 lis 2022 · Section 6.4 : Volume With Cylinders. For each of the following problems use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis. Rotate the region bounded by \(x = {\left( {y - 2} \right)^2}\), the \(x\)-axis and the \(y\)-axis about the \(x\)-axis. Solution
18 lip 2024 · To calculate the volume of a hollow cylinder, you can use a rough but effective formula and subtract the volume of the two right cylinders: V_ {\text {H}} = V_ {\text {C}_1}-V_ {\text {C}_2} VH=VC1−VC2. where: V H V_ {\text {H}} VH — Volume of the hollow cylinder; V C 1 V_ {\text {C}_1} VC1 — Volume of the bigger, outer cylinder; and.
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6.3.1 Calculate the volume of a solid of revolution by using the method of cylindrical shells. 6.3.2 Compare the different methods for calculating a volume of revolution.
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Fortunately, there is a method, called the method of cylindrical shells, that is easier to use in such a case. Figure 2 shows a cylindrical shell with inner radius , outer radius , and height . Its volume is calculated by subtracting the volume of the inner cylinder from the volume of the outer cylinder:
