Approximate area beneath a curve from ​x=a​ to ​x=b​ by summing the area of ​n​ trapezoids of equivalent width.

Each trapezoid's area is the common width ​(nb−a​)​ times the average of the function's value on the left and rigth side of the trapezoid:

Integration, or anti-differentiation, is essentially the opposite of differentiation. We use the integral symbol​∫​ and write:

By convention we denote this function ​F:

We can also write

Notice the ​dx​ under the integral. This tells us which variable we are integrating with respect to - in this case we are reversing ​dxd​.

Since the derivative of a constant is always zero, then if if ​F′(x)=f(x), then ​(F(x)+C)′=f(x).

This means that when we integrate, we can add any constant to our result, since differentiating makes this constant irrelevant:

In the same way that constant multiples can pass through the derivative, they can pass through the integral:

And in the same way that the derivative of a sum is the sum of the derivatives:

If we know the value of ​y​ or ​f(x)​ for a given ​x, we can determine ​C​ by plugging in ​x​ and ​y.

A definite integral is evaluated between a lower and upper bound.

We can solve a definite integral with

where ​F(x)=∫f(x)dx.

Graphing calculators can be used to evaluate definite integrals.

For example, on a TI-84, math > 9:fnInt(, which prompts you with ​∫□□​(□)d□. Make sure the variable of your function matches the variable that you take the integral with respect to.

Integrals of the same function with adjacent domains can be merged:

Similarly, the domain of an integral can be split:

for any ​a<m<b.

Approximate area beneath a curve from ​x=a​ to ​x=b​ by summing the area of ​n​ trapezoids of equivalent width.

Each trapezoid's area is the common width ​(nb−a​)​ times the average of the function's value on the left and rigth side of the trapezoid:

Integration, or anti-differentiation, is essentially the opposite of differentiation. We use the integral symbol​∫​ and write:

By convention we denote this function ​F:

We can also write

Notice the ​dx​ under the integral. This tells us which variable we are integrating with respect to - in this case we are reversing ​dxd​.

Since the derivative of a constant is always zero, then if if ​F′(x)=f(x), then ​(F(x)+C)′=f(x).

This means that when we integrate, we can add any constant to our result, since differentiating makes this constant irrelevant:

In the same way that constant multiples can pass through the derivative, they can pass through the integral:

And in the same way that the derivative of a sum is the sum of the derivatives:

If we know the value of ​y​ or ​f(x)​ for a given ​x, we can determine ​C​ by plugging in ​x​ and ​y.

A definite integral is evaluated between a lower and upper bound.

We can solve a definite integral with

where ​F(x)=∫f(x)dx.

Graphing calculators can be used to evaluate definite integrals.

For example, on a TI-84, math > 9:fnInt(, which prompts you with ​∫□□​(□)d□. Make sure the variable of your function matches the variable that you take the integral with respect to.

Integrals of the same function with adjacent domains can be merged:

Similarly, the domain of an integral can be split:

for any ​a<m<b.