8.2 Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration. It shows that these seemingly opposite operations are actually closely related, allowing us to find definite integrals using antiderivatives.

This theorem is a game-changer in calculus. It gives us a powerful tool to calculate areas under curves and solve complex integration problems, making it easier to tackle real-world applications in physics, engineering, and economics.

Fundamental Theorems of Calculus

First Fundamental Theorem of Calculus

Top images from around the web for First Fundamental Theorem of Calculus

• 

  The Fundamental Theorem of Calculus · Calculus View original

  Is this image relevant?

• 

  Fundamental theorem of calculus - Wikipedia View original

  Is this image relevant?

• 

  Understanding Integration, Integrals, Antiderivatives, and their Relationship to Derivatives ... View original

  Is this image relevant?

• 

  The Fundamental Theorem of Calculus · Calculus View original

  Is this image relevant?

• 

  Fundamental theorem of calculus - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for First Fundamental Theorem of Calculus

• 

  The Fundamental Theorem of Calculus · Calculus View original

  Is this image relevant?

• 

  Fundamental theorem of calculus - Wikipedia View original

  Is this image relevant?

• 

  Understanding Integration, Integrals, Antiderivatives, and their Relationship to Derivatives ... View original

  Is this image relevant?

• 

  The Fundamental Theorem of Calculus · Calculus View original

  Is this image relevant?

• 

  Fundamental theorem of calculus - Wikipedia View original

  Is this image relevant?

1 of 3

• States that if $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$ on $[a,b]$, then \int_{a}^{b} f(x) dx = [F(b) - F(a)](https://www.fiveableKeyTerm:f(b)_-_f(a))
• Establishes a connection between the definite integral and the antiderivative
• Allows for the calculation of definite integrals using antiderivatives
• Provides a way to evaluate the area under a curve by finding the difference between the antiderivative values at the endpoints of the interval
• Example: If $f(x) = x^2$ and $F(x) = \frac{1}{3}x^3 + C$ is an antiderivative of $f$, then $\int_{1}^{2} x^2 dx = F(2) - F(1) = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$

Second Fundamental Theorem of Calculus

• States that if $f$ is continuous on $[a,b]$, then $\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$ for all $x$ in $[a,b]$
• Establishes a connection between the definite integral and the derivative
• Allows for the differentiation of definite integrals with respect to the upper limit of integration
• Provides a way to find the derivative of a function defined as a definite integral
• Example: If $F(x) = \int_{0}^{x} \sin(t) dt$, then $F'(x) = \frac{d}{dx} \int_{0}^{x} \sin(t) dt = \sin(x)$

Evaluation of Definite Integrals

• The Fundamental Theorems of Calculus provide methods for evaluating definite integrals
• The First Fundamental Theorem allows for the calculation of definite integrals using antiderivatives
• The Second Fundamental Theorem allows for the differentiation of functions defined as definite integrals
• Definite integrals can be evaluated by finding an antiderivative, substituting the limits of integration, and subtracting the results
• Example: To evaluate $\int_{0}^{1} e^x dx$, find an antiderivative $F(x) = e^x$, then calculate $F(1) - F(0) = e - 1$

Antiderivatives and Integral Functions

Antiderivatives

• An antiderivative of a function $f$ is a function $F$ whose derivative is $f$, that is, $F'(x) = f(x)$
• Antiderivatives are not unique; if $F$ is an antiderivative of $f$, then $F + C$ is also an antiderivative of $f$ for any constant $C$
• The process of finding an antiderivative is called antidifferentiation or indefinite integration
• The set of all antiderivatives of a function is called the indefinite integral of the function
• Example: If $f(x) = x^2$, then $F(x) = \frac{1}{3}x^3 + C$ is an antiderivative of $f$ for any constant $C$

Integral Functions

• An integral function is a function defined by a definite integral with a variable upper limit of integration
• The integral function $F(x) = \int_{a}^{x} f(t) dt$ represents the area under the curve of $f$ from $a$ to $x$
• The Second Fundamental Theorem of Calculus states that if $f$ is continuous on $[a,b]$, then the integral function $F(x) = \int_{a}^{x} f(t) dt$ is continuous on $[a,b]$ and differentiable on $(a,b)$, with $F'(x) = f(x)$
• Example: The integral function $F(x) = \int_{0}^{x} \cos(t) dt$ represents the area under the curve of $\cos(t)$ from $0$ to $x$

Continuous Functions

• A function $f$ is continuous on an interval $[a,b]$ if it is continuous at every point in the interval
• Continuity is a necessary condition for the Fundamental Theorems of Calculus to hold
• If a function is continuous on an interval, then it is integrable on that interval
• The sum, difference, product, and quotient (when the denominator is not zero) of continuous functions are also continuous
• Example: The function $f(x) = x^2$ is continuous on the interval $[0,1]$, so it is integrable on that interval, and the Fundamental Theorems of Calculus can be applied to it