Calculus Notes

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The Shell Method

The shell method of finding the volume of solids of revolution

All integrals are preceded by 2 pi to account for the area of the circular base

Integrate the entire shaded region of the graph

The Integral will have the two parts

The length measured from the axis of rotation

The height from the perpendicular axis.

If the axis of rotation is the x-axis, integrate in terms of y

If the axis of rotation is the y-axis, integrate in terms of x

This method does not work when you hit the same function twice in the process of drawing your rectangle

If you hit different functions twice, then think about subtracting the pieces to get what you need. You only have to do this once.

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Fluid Pressure

Finding fluid pressure

Integrals are preceded by the weight of whatever fluid is pressing on them.

Draw the situation in such a space as makes it easy

Integrate from the height of the surface

Integrate the equation describing the width of the surface as well as the height as it changes.

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Area Between Two Functions

Finding the area between two functions

Integrate from intersection to intersection

The function that is on top goes first

The function of bottom is subtracted from the first

If the positions of the function change, split up the integral

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Integrating Exponential Functions

Integrating exponential functions

E to the x integrates to E to the x plus c

One can use u substitution on the exponent of e as well

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Disc and Washer Methods

Disc Method

Integrate the shaded region along the axis that you are revolving around

All integrals are preceded by pi

Find the radius of the disc and then square it

Make sure that all the variables are the same and that you are integrating along that axis

Washer Method

Same as above

Find the bigger radius and square it

Find the smaller radius and square it

Subtract the smaller radius from the larger radius inside of the integral

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Spring Problems

Remember: Force equals spring constant times distance

Use the information given to find k

Integrate from wherever you begin stretching to wherever you finish stretching from the equilibrium point

Keep the same units throughout

Integrate the spring constant times x

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Arc Length

Integrate arc length from whatever x coordinates

Integrate the square root of the quantity of one plus the derivative of the function squared

Surface Area

Preceded by two pi

Integrate from whatever x coordinates of the thing being rotated

Integrate the radius multiplied by the arc length

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Emptying Problems

Draw a model in such a place as makes the problem easiest

Integrate the area of the cross section

Integrate from the top coordinate of the thing being emptied to the bottom

Multiply the area formula by the height from which the container is being emptied subtracted by y

Integrate in terms of y

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Integrating Fractions

"One over" fractions integrate into natural logarithms

Make sure to use U substitution

Sometimes long devision, factoring, completing the square, and other strategies might be necessary for getting the numerator under control

Exponential Functions

Treat the exponential function as you would any other system of equations

Use logarithms to find the constant of proportionality

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Newton's Law of Cooling

Newton's Law of cooling states that the rate an object cools at is proportional to the difference between the temperature of the object and the surrounding temperature.

The form of the equation is T=Ce^kt +Ta(the ambient temperature)

C equals the difference between the initial temperature of the object and the ambient temperature

Use the given point to solve for k

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Integrating by Parts

Integrated form of the Product rule

uv= the integral of udv plus the integral of vdu

Hence, the integral of udv=uv minus the integral of vdu

To integrate by parts, find u and dv

dv should be the most complicated term you can integrate

u should be everything left over

Integrate dv to find v

derive u to find du

Plug into the equation and solve.

x^n, lnx, arcsinx, arctanx are usually u's

e^x, sinx, cosx, are usually dv's

The Tabular Method

Sometimes you will have to do separation of variables multiple times, which is long and annoying.

In the Tabular method, find u and dv as normal

Then derive u, entering the result in a table, until you reach zero

Then integrate dv, entering the result in a table, until you match the u's

Then plug the results into the pattern of uv-u1v1+u2v2... etc.

The v will e one cell down from the corresponding u