The Integral Part of Calculus

The Integral Part of CalculusDue to the importance of calculus, it is essential to understand its fundamentals. This discusses the second fundamental theorem of calculus and its derivation. This theorem allows us to compute integrals and is used often in many classes and careers.

Previously, we proved the first fundamental theorem of calculus (FTC). There still remains the second part of this theorem. The second FTC states that \( \int_{a}^{b}f(x)dx=F(b)-F(a) \). Since the first FTC explains that differentiating the integral gives back the function in the integrand, the integral must be equal to the antiderivative of the function \( F(x)(\frac{\mathrm{d} }{\mathrm{d} x}F(x)=f(x)) \). In other words, the indefinite integral \( \int_{}^{}f(x)=F(x)+C \) where \( C \) is an arbitrary constant. Since \( \frac{\mathrm{d} }{\mathrm{d} x}F(x)=\frac{\mathrm{d} }{\mathrm{d} x}F(x)+5=\frac{\mathrm{d} }{\mathrm{d} x}F(x)-5=f(x) \), any real constant (positive or negative) added to the antiderivative will still equal \( f(x) \) when the full function is differentiated. For this reason, the indefinite integral represents a family of functions because all real values of \( C \) satisfy the first FTC. This arbitrary constant will be important in the following proof.

While proving the first FTC, we used the function \( A(x)=\int_{a}^{x}f(t)dt \) to represent the area under \( f(x) \) from some constant \( a \) to a variable \( x \). With this, the second FTC can be represented as \( A(b) \) because \( A(b)=\int_{a}^{b}f(x)dx \) which equals \( F(b)+C \) due to the previous definition of the integral. To solve this, we must first solve \( A(a)=\int_{a}^{a}f(x)dx=F(a)+C \), however, because the width between \( a \) and \( a \) is 0, we know the area must be 0 as well so \( A(a)=0=F(a)+C \). Rearranging this, we get \( C=-F(a) \). This means that \( \int_{a}^{b}f(x)dx=F(b)-F(a) \) which is the second FTC.


Shlok Bhattacharya

Leave a comment

Share This:


Leave a Reply

Your email address will not be published. Required fields are marked *

x Logo: Shield Security
This Site Is Protected By
Shield Security