Calculating the area under a curve is an important idea in calculus, especially when we use something called integration.
In simple terms, if we want to find the area under the curve of a function ( f(x) ) from point ( a ) to point ( b ), we can use the definite integral, which looks like this:
[ \text{Area} = \int_{a}^{b} f(x) , dx ]
Identify the Function: First, figure out the function ( f(x) ) that you want to find the area for.
Set Limits: Next, choose the starting point ( a ) and the ending point ( b ).
Integrate: Now, you need to calculate the definite integral using something called antiderivatives.
Evaluate: Finally, subtract the values of the antiderivative at the endpoints:
[ \text{Area} = F(b) - F(a) ]
Let’s say we have the function ( f(x) = x^2 ) and we want to find the area between ( x = 1 ) and ( x = 3 ):
[ \text{Area} = \int_{1}^{3} x^2 , dx = \left[\frac{x^3}{3}\right]_{1}^{3} ]
Now we calculate it:
[ = \left(\frac{27}{3} - \frac{1}{3}\right) = \frac{26}{3} ]
This means that the total area under the curve from ( x = 1 ) to ( x = 3 ) is ( \frac{26}{3} ).
Calculating the area under a curve is an important idea in calculus, especially when we use something called integration.
In simple terms, if we want to find the area under the curve of a function ( f(x) ) from point ( a ) to point ( b ), we can use the definite integral, which looks like this:
[ \text{Area} = \int_{a}^{b} f(x) , dx ]
Identify the Function: First, figure out the function ( f(x) ) that you want to find the area for.
Set Limits: Next, choose the starting point ( a ) and the ending point ( b ).
Integrate: Now, you need to calculate the definite integral using something called antiderivatives.
Evaluate: Finally, subtract the values of the antiderivative at the endpoints:
[ \text{Area} = F(b) - F(a) ]
Let’s say we have the function ( f(x) = x^2 ) and we want to find the area between ( x = 1 ) and ( x = 3 ):
[ \text{Area} = \int_{1}^{3} x^2 , dx = \left[\frac{x^3}{3}\right]_{1}^{3} ]
Now we calculate it:
[ = \left(\frac{27}{3} - \frac{1}{3}\right) = \frac{26}{3} ]
This means that the total area under the curve from ( x = 1 ) to ( x = 3 ) is ( \frac{26}{3} ).