CALCULUS II EXAM I NOTES AND LINKS

CALCULUS II EXAM I NOTES AND LINKS

Create Your Formula Sheet In Advance

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this web site.

QuickTime free download.

	THE INTEGRATOR is a free function integrator. You need to read their instructions on entering input to use it effectively. It is from Wolfram Research and powered by Mathematica.
	Quickmath This is a free online source for solving math problems powered by Mathematica.
	You can also compute integrals using Derive on your computer in class to check your answers or you can use the antiderivative feature on the Vanderbilt Toolkit. The Vanderbilt Toolkit includes some specialized utilities that relate directly to chapter 6. Regrettably, this free online software has not been available for quite a while now.
	Hotmath You can look at solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, sixth , seventh, and eighth editions. They have chapters 1 - 12 available. Only a few solutions are still free (solutions to problems 15, 25, 35 in each section are free).
	Traces Animes A really nice set of 2D and 3D WIMS graphing tools by XIAO Gang which will produce pictures and graphs that can then be saved and printed or used on a web site. Traces Animes sample animation
	Printable worksheets for graphical exercises can be found at mathgraphs.com.
	Visual Calculus has lots of great stuff.
	Textbook support site
	Here is a Maple Worksheet with integration (and differentiation) examples.
	Here is a Maple Worksheet on differentiating and integrating logarithmic and exponential functions.
	Here is a preview of the area applications in Calculus II including area between two curves.

5.6 You will need to be able to define and differentiate the inverse trigonometric functions. Here is a presentation of the inverse tangent, inverse sine, and inverse cosine from the UBC online calculus course.

y = arcsin(x) y = arctan(x) y = arcsec(x)

5.7 You will need to be able to use the inverse trigonometric functions in integration. You will need to complete the square in computing some of the integrals in this section.

5.8 You will need to be able to define the hyperbolic functions and their inverses and use them in differentiation and integration.

y = arcsinh(x) y = arccosh(x) y = arctanh(x)

6.1-6.3 We will review solving a few differential equations and their applications.

Hopefully in Calculus I you learned to solve some ordinary differential equations (ODES) by the method of separation of variables. To have a better sense of what the solution to a first order ODE is take a look at this First Order DE Solution Grapher. The picture on the right shows a blow-up of a portion of what the DE Solution Grapher will produce, in this case showing part of the graph of the solution to dy/dx = 2x, y(0) = 0. The blue line segments indicate the direction of tangents to the graph of any solution to dy/dx = 2x that went through the left endpoint of the line segment.

Try using the method of separation of variables to solve the following:

dy/dx = y(cos(x)), y(0) = 1 The solution can be found in the examples.

The graph of the solution is shown on top at the right. Click on the graph to see an animation of the direction field vectors moving across the screen for increasing values of x along with an animated solution point. The lower graph on the right shows the solutions for y(0) = -1, -0.4, 0.4, and 1 along with the direction field. Click on the graph to see an enlargement.

In addition to the graphs shown at the right you can look at various solutions corresponding to an initial condition of the form y(0) = c by following this link to a DPGraph of a surface and the plane y = c. The curve of intersection of the surface and the plane when c = 1 is the graph of the solution to the initial value problem above. You can use the scrollbar and activate c to look at solutions for various values of c. You can also use the z-slice feature.

Here is another java applet DE solution grapher that also draws the direction field.

Here are more examples of solutions to first order separable differential equations from my Differential Equations course web site.

7.1 You will need to be able to find the area of a region between two curves. Here is the SOS Math introduction to the area between two curves and an applet demonstrating area between two curves. Here are some drill problems on area between two curves. This is an animation reviewing approximating the area under part of a sine wave.

7.2 You will need to be able to compute volumes using the disk method. You should also be able to compute the volume of a solid whose cross section area function is known (See Page 461 and these two Quicktime movies by Bruce Simmons: The Concept, An Example). Another example of this would be a solid whose bottom is bounded by a circle and whose cross sections are squares. Here is a java applet illustrating such a problem.

Here is the surface formed by revolving the graph of y = (x³ - 2x² - 3x + 10)/10 about the x-axis over the interval [-2,2].

This DPGraph picture is of the Surface of Revolution formed by revolving the graph of y = 1 + (x+4)(x-4)((x-1)/20 about the x-axis over the interval [-3,3].

This DPGraph picture is of the Surface of Revolution 2 formed by revolving the graph of y = 3 + (2 + 8sin(x))/(1.2x + 1) about the x-axis over the interval [0,18].

7.3 You will need to be able to compute volumes using the shell method.

Here is the HMC tutorial on finding volumes and the UBC tutorial on finding volumes.

7.4 You will need to be able to compute arc length and the area of a surface of revolution. Arc Length1 This utility will compute the arc length for a function f(x) over an interval [a,b]. Arc Length2 This is a nice applet demonstrating the approximation of arc length using line segments. Solid of Revolution 1 This utility will draw the surface formed by rotating the graph of y=f(x) over [a,b] about the x-axis or y-axis. Solid of Revolution 2 This utility will draw the solid formed by rotating the intersection of the graphs of y=f(x) and y=g(x) over [a,b] about the x-axis or y-axis. Another nice arc length applet is found in the HMC arclength tutorial. Below are two pictures demonstrating the approximation of arc length along the graph of the parabola y = x² from x = -2 to x = 2. The figure on the left uses 4 line segments and the figure on the right uses 8.

7.5 You will need to be able to compute the work done when the force being applied is variable. Examples of this would be the work done in compressing or stretching a spring (using Hooke's Law), emptying the liquid out of a tank, and lifting a chain. Look at this animation (see picture on the left) and see if you can figure out how much work is done against gravity in lifting the chain and the weight attached to the end of it up to the ceiling. The chain weighs one pound per foot. After the "oops" the weight comes loose and falls back to the floor. How fast is the weight traveling (neglecting air resistance) when it hits the floor? Here is the UBC tutorial on work.

Spring Stretching Example

A force of 100 pounds will stretch a spring 2 feet beyond its equilibrium length of 5 feet. Find the work done in stretching the spring from a length of 5 feet to a length of 8 feet. Click here to see an animation and click here to see an animation with scales.

100 = 2k so the spring constant is 50. The work done would be

How much additional work would be done in stretching the spring two more feet (assuming we are still within the elastic limits of the spring and Hooke's Law holds)?

7.6 You will need to be able to find the moments and center of mass of a planar lamina with uniform density. You will need to be able to find the centroid of a plane region. Center of Mass This utility will compute the center of mass of a plate with uniform density whose boundaries are y=f(x) (on top), y=g(x) (on the bottom), x=a, and x=b.

7.7 You will need to be able to find the total fluid force on one side of a vertical submerged surface. We will also look at how to find the total fluid force on one side of a submerged flat surface that is not vertical.

TAKE HOME PROBLEMS Due May 31

1. Find the center of mass of the planar lamina whose boundaries are given by y = x² and y = x + 2 with constant density 2 units per square unit. Click here to see a picture. DPGraph Picture

2. A rectangular box shaped container of water has dimensions 10' by 2' by 10' (see picture). It is filled with water to a depth of 4 feet (again see picture). Using a large tube the water is to be pumped out of the box. In doing so all the water will need to be pumped to a height that is 4 feet above the top of the box after which it will fall to the ground. Compute the total work done against gravity in pumping all the water to a height 4 feet above the top of the box. Click on either picture below to see an animation of what is happening.

3. A horse watering trough is filled with water to a depth of 2 feet. The trough is in the shape of a supported hemisphere with a diameter across the top of 6 feet. (See a picture below on the left.) How much work against gravity is required to pump all the water to the top of the trough? The density of water is 62.4 pounds per cubic foot. Sketch the picture, model the problem in terms of a series, and give and compute the integral.

4. A plate in the shape of a right triangle with legs of length 2 feet and 3 feet is submerged in a pool of water. The 2 foot long leg is resting on the bottom of the pool parallel to the surface of the water. The plate is vertical and the pool is 10 feet deep where the plate is. (See a picture above on the right.) What is the total fluid force of the water on one side of the plate? Sketch the picture, model the problem in a series, give the integral and evaluate the integral.

Here is a large picture of the vase and the data for the volume and surface area extra credit problem.

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Lane Vosbury, Math Chair, Seminole Community College email: vosburyl@scc-fl.edu