Fundamental+Theorem

​​​Fundamental Theorem of Calculus

The fundamental theorem of calculus is the formula used to find the area under a curve or function from one x-value to another x-value. For some functions, you can easily find the area under the curve without using the fundamental theorem, if the two x-values are small and close together. Take for instance y = x:



If you wanted to find the area under the curve of y = x from x=0 to x = 5, you could calculate it without the theorem. The area under y = x from 0 - 5 creates a triangle with base 5 and height 5. Using the formula for area of a triangle (A = 1/2 * b * h), you can calculate that the area under the curve from 0 - 5 is 1/2 * 5 * 5, or 12.5. You can also think of the area as half of a square with sides equal to 5. The area would be 1/2 * 25, also equalling 12.5. You can also use the fundamental theorem of calculus and will need it to calculate more difficult areas.

The Fundamental Theorem of Calculus is defined in the textbook like this -

If //f// is an integrable function and if g(x) = integral of f(x)dx, then:

= g(b) - g(a)

Now, take the example above, the area under the curve of y = x from x = 0 to x = 5, and apply it to the theorem.

g(x) = integral of x (dx), integrated from 0 to 5. The integral looks like this:



The integral of x is 1/2 x 2, so g(x) = 1/2 x 2.

So, g(b) - g(a) is just 1/2 (5) 2 - 1/2 (0), which equals 12.5 - 0, or just 12.5 (the same value calculated above).

Now, as shown before, you can sometimes calculate the area under a curve without using the fundamental theorem, but for some functions, you need the theorem.

Example 1: y = e x.



If you want to find the area under the curve of y = e x from for instance, -2 to .5, you can use the fundamental theorem of calculus:

The integral of e x is e x, so g(x) is e x.

g(.5) - g(-2) = e(.5) - e(-2) = 1.511.5

Example 2: y = sin(x) To find the area under the curve of y = sin x from say, x=2 to x= 7, then you can integrate using the fundamental theorem.

The integral of sin x is - cos x, so g(x) = - cos x.

g(7) - g(2) = - cos (7) - - cos(2) = -1.17

Here's a good video that goes through some harder problems with the theorem:

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