To use the Fundamental Theorem of Calculus (FTC) with trigonometric functions, you can follow these simple steps:
Identify the Function:
Start with a function like ( f(x) = \sin(x) ) or ( f(x) = \cos(x) ).
Find the Antiderivative:
The antiderivative helps us find the area under the curve:
Evaluate the Definite Integral:
To find the area under the curve between two points ( a ) and ( b ), use the FTC:
[
\int_a^b f(x) , dx = F(b) - F(a)
]
Here, ( F(x) ) is the antiderivative.
Example Calculation:
Let’s look at an example using ( f(x) = \sin(x) ) from ( 0 ) to ( \pi ):
[
\int_0^\pi \sin(x) , dx = [-\cos(x)]_0^\pi = -\cos(\pi) - (-\cos(0)) = 2.
]
This shows how to use the FTC to easily calculate the area under trigonometric curves!
To use the Fundamental Theorem of Calculus (FTC) with trigonometric functions, you can follow these simple steps:
Identify the Function:
Start with a function like ( f(x) = \sin(x) ) or ( f(x) = \cos(x) ).
Find the Antiderivative:
The antiderivative helps us find the area under the curve:
Evaluate the Definite Integral:
To find the area under the curve between two points ( a ) and ( b ), use the FTC:
[
\int_a^b f(x) , dx = F(b) - F(a)
]
Here, ( F(x) ) is the antiderivative.
Example Calculation:
Let’s look at an example using ( f(x) = \sin(x) ) from ( 0 ) to ( \pi ):
[
\int_0^\pi \sin(x) , dx = [-\cos(x)]_0^\pi = -\cos(\pi) - (-\cos(0)) = 2.
]
This shows how to use the FTC to easily calculate the area under trigonometric curves!