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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

At a Glance - Order of Limits of Integration

Integrals like to flip-flop on their stance from time to time. Seriously, they're as bad as politicians sometimes. Sometimes you think they're left, sometimes you think they're right, sometime the upper limit is smaller than the lower limit...

When we originally stated the FTC we said that if f is continuous on [a, b], then

where F ' = f.

We can still evaluate integrals this way if the upper limit of integration is smaller than the lower limit.

Suppose this is the case, so b < a. By properties of integrals,

Since b < a we can use the FTC to say

Then

Practically speaking, this means you can evaluate integrals without worrying which limit of integration is bigger. The integrand should still be continuous on the interval between the limits of integration, though.

Example 1

Evaluate .


Exercise 1

Evaluate the integral.


Exercise 2

Evaluate the integral.


Exercise 3

Evaluate the integral.


Exercise 4

Evaluate the integral.


Exercise 5

Evaluate the integral.


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