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- Determine location as a function of time using integration and dynamic motion.
- Use root finding and integration to calculate simple areas.
- Calculate the area between two functions using integration.
- Use single integrals to calculate volumes.
- Determine volumes of revolution using integration.
1. Integration and Dynamic Motion
This lesson uses driving to demonstrate how graphing can help you use calculus to figure out how fast you were going at a given time. Using this lesson, you can learn how to integrate your velocity to find your position.
2. How to Find Simple Areas With Root Finding and Integration
Combine your calculus tools in this lesson! Find the area enclosed by multiple arbitrary functions by first finding the root of an equation and then integrating over the resulting range.
3. How to Find Area Between Functions With Integration
Sometimes you aren't looking for the area under the curve; after all, not every region is between a curve and axis! In this lesson, learn how to find areas between curves as well as areas with complicated boundaries.
4. How to Calculate Volumes Using Single Integrals
Ever wonder where the equation for the volume of a cone comes from? Or the equation for the volume of a sphere? In this lesson, learn how to use a slicing technique to find the volume of a region by solving a single integral.
5. How to Find Volumes of Revolution With Integration
Some shapes look the same as you rotate them, like the body of a football. In this lesson, learn how to find the volumes of shapes that have symmetry around an axis using the volume of revolution integration technique.
6. Solving the Integral of cos(2x)
In this lesson, we will find the integral of cos(2x) using integration by substitution in a step-by-step process. We will then go on to check our work using derivatives and the chain rule for derivatives.
7. Solving the Integral of cos(x)
In this lesson, see how to find the integral of cos(x) using well-known derivatives and the fundamental theorem of calculus. We also examine how we can use this theorem to see a connection between derivatives and integrals.
8. Solving the Integral of ln(x)
Finding the integral of ln(x) requires the process of integration by parts. This lesson will show how to find the integral of ln(x) using integration by parts and explain how and why integration by parts works.
9. Finding the Antiderivative of 1/cos(x)
It's handy to know the antiderivatives of certain functions. In this lesson, we will calculate the antiderivative of 1 / cos(x). First, we'll use a trigonometric identity to see how to use a well-known formula to find this antiderivative, and then we will take a look at where this formula comes from.
10. Finding the Integral of e^x
In this lesson, we show how to find the integral of e^x by considering the series expansion for the exponential function and the fundamental theorem of calculus. A numerical application is included which highlights the integration result.
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