Would someone clarify this for me... Is antiderivatives just another name for intergral and why is intergral of a function is the area under the curve?
Antiderivatives is another word for integral.
Why area? THink on this...the integral is the sum of all the "heights" times width (dx). Take a look at this depiction of the trapezoidal rule:
http://metric.ma.ic.ac.uk/integration/techniques/definite/numerical-methods/trapezoidal-rule/
The anti-derivative is another name for the indefinite integral. The term integral is ambiguous here, we have both definite and indefinite integrals.
The definite integral is the area bounded by the function, the x-axis and the limits of integration. The last part of your question is slightly confusing. The definite integral is defined as the limit of a sum which represents the area under the function.
To clarify, the term "antiderivative" refers to the process of finding an expression that, when differentiated, would yield the original function. This is often denoted as ∫f(x)dx, where f(x) is the original function and dx represents the variable of integration. In other words, the antiderivative is the reverse of differentiation.
Now, in calculus, the integral is divided into two main types: the definite integral and the indefinite integral. The definite integral calculates the area between a function and the x-axis within specific boundaries (limits of integration). It is represented mathematically as ∫f(x)dx evaluated from a to b. The result of a definite integral is a numerical value representing the area.
On the other hand, the indefinite integral (also known as the antiderivative) represents a family of functions that, when differentiated, would yield the original function. It does not involve specific boundaries but rather a constant of integration, often denoted as + C. The result of an indefinite integral is a general expression rather than a specific numerical value.
As for why the definite integral calculates the area under the curve, it can be understood by considering the concept of the Riemann Sum. This method approximates the area by dividing it into small rectangles or trapezoids. As the width of these shapes approaches zero, the sum of their areas becomes closer to the actual area under the curve. Taking the limit of this sum as the width approaches zero (which is the definition of the definite integral) gives us the exact area.
In summary, the terms "antiderivative" and "integral" are related in that they both involve finding a function associated with another function. However, they have different interpretations: the antiderivative represents a family of functions, while the integral calculates the area under a curve.