3.1.6 AP calculus Exam piece wise integral

In summary, a piece wise integral is an integral that is computed over different intervals of a function, where each interval has a different expression or formula for the function. It allows for more flexibility in computing integrals for complex functions. This concept is tested in AP calculus exams to assess a student's understanding and application of integrals in different scenarios. To solve a piece wise integral problem, one must identify the different intervals and corresponding functions, compute the integral separately for each interval, and combine the results. An example of a piece wise integral problem is finding the area under a curve with different intervals for x values.
  • #1
karush
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I tried to do this just by observation, but kinda hard with a piece wise function
so would presume

$\displaystyle \int_1^3 2 \, dx +\int_3^5 x-1 \, dx
= 2x\biggr|_1^3 + \left(\dfrac{x^2}{2}-x\right)\biggr|_3^5=4+6=10$

i wasn't sure about the notation of limits when you have an inequality in the function
 

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  • #2
that’s fine
 
  • #3
karush said:
I tried to do this just by observation, but kinda hard with a piece wise function
so would presume

Keep in mind that you could have solved this one by Plane Geometry; no calculus required. Is this what you meant with "by observation"?

Clearly, it's okay to use the calculus in your calculus class. Just keep it in mind. :-)
 
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Related to 3.1.6 AP calculus Exam piece wise integral

1. What is a piecewise integral in AP Calculus?

A piecewise integral is an integral that involves multiple functions with different intervals of integration. This is commonly seen in piecewise-defined functions, where the function changes at a specific point or interval.

2. How do you solve a piecewise integral?

To solve a piecewise integral, you need to break it down into smaller integrals based on the different intervals. Then, you can use the appropriate integration technique for each interval, such as u-substitution or integration by parts.

3. What is the difference between a definite and indefinite piecewise integral?

A definite piecewise integral has specific limits of integration, while an indefinite piecewise integral does not. This means that a definite integral will give you a numerical value, while an indefinite integral will give you a function.

4. Can you use the Fundamental Theorem of Calculus to solve a piecewise integral?

Yes, the Fundamental Theorem of Calculus can be used to solve a piecewise integral. This theorem states that the derivative of an integral is equal to the original function, and vice versa. It can be applied to both definite and indefinite piecewise integrals.

5. Are there any tips for solving piecewise integrals?

One helpful tip for solving piecewise integrals is to carefully identify the different intervals and functions involved. It can also be useful to graph the piecewise function to better visualize the intervals and understand the behavior of the function. Additionally, practice and familiarity with integration techniques can make solving piecewise integrals easier.

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