The Fundamental Theorem of Calculus (FTC) is an important idea that links two big topics in math: differentiation and integration. It helps us figure out areas and volumes in a smooth way.
The FTC says that if we have a function ( f(x) ) that is continuous (this means it doesn't jump around) on the interval from ( a ) to ( b ), then:
We can find a new function ( F(x) ) that represents the area under the curve of ( f(x) ) from ( a ) to any point ( x ). This new function is called an antiderivative.
We can use this antiderivative to find the area under the curve from ( a ) to ( b ) using the formula:
( \int_{a}^{b} f(x) , dx = F(b) - F(a) )
When we want to find the area under a curve, we can think of the definite integral as the space between the curve ( y = f(x) ) and the x-axis from ( x = a ) to ( x = b ).
For example, if we want to find the area under the curve ( y = x^2 ) from ( x = 1 ) to ( x = 3 ), we can write it like this:
( \int_{1}^{3} x^2 , dx )
This gives us the area between the curve and the x-axis.
The FTC also helps us find volumes, especially when we spin a shape around an axis. When we turn a region around an axis, we can use integration to calculate the volume.
For example, the volume ( V ) of a solid formed by rotating the curve ( y = f(x) ) around the x-axis from ( x = a ) to ( x = b ) can be found with this formula:
( V = \pi \int_{a}^{b} [f(x)]^2 , dx )
This shows us how the FTC helps us find both areas and volumes, connecting shapes we see in geometry with the tools of calculus.
The Fundamental Theorem of Calculus (FTC) is an important idea that links two big topics in math: differentiation and integration. It helps us figure out areas and volumes in a smooth way.
The FTC says that if we have a function ( f(x) ) that is continuous (this means it doesn't jump around) on the interval from ( a ) to ( b ), then:
We can find a new function ( F(x) ) that represents the area under the curve of ( f(x) ) from ( a ) to any point ( x ). This new function is called an antiderivative.
We can use this antiderivative to find the area under the curve from ( a ) to ( b ) using the formula:
( \int_{a}^{b} f(x) , dx = F(b) - F(a) )
When we want to find the area under a curve, we can think of the definite integral as the space between the curve ( y = f(x) ) and the x-axis from ( x = a ) to ( x = b ).
For example, if we want to find the area under the curve ( y = x^2 ) from ( x = 1 ) to ( x = 3 ), we can write it like this:
( \int_{1}^{3} x^2 , dx )
This gives us the area between the curve and the x-axis.
The FTC also helps us find volumes, especially when we spin a shape around an axis. When we turn a region around an axis, we can use integration to calculate the volume.
For example, the volume ( V ) of a solid formed by rotating the curve ( y = f(x) ) around the x-axis from ( x = a ) to ( x = b ) can be found with this formula:
( V = \pi \int_{a}^{b} [f(x)]^2 , dx )
This shows us how the FTC helps us find both areas and volumes, connecting shapes we see in geometry with the tools of calculus.