Fundamental Theorem of Calculus (AQA A-Level Mathematics): Revision Notes

Part 1: The Integral as an Antiderivative (FTC1):

If $f(x)$ is continuous on a closed interval $[a, b]$, then the function $\ F(x)$ defined by: $\ F(x) = \int_a^x f(t) \, dt$ is continuous on$\ [a, b]$, differentiable on the open interval $\ (a, b),$ and its derivative is $\ f(x)$. In other words: $F'(x) = \frac{d}{dx} \left(\int_a^x f(t) \, dt\right) = f(x)$

Interpretation: This part of the theorem tells us that if we integrate a function $\ f(x)$from $\ a$ to $\ x$, and then differentiate the result, we get back the original function $\ f(x)$.

Part 2: The Evaluation of Definite Integrals (FTC2):

If $\ f(x)$ is continuous on the closed interval $\ [a, b]$ and $F(x)$ is any antiderivative of $\ f(x) \ (i.e., \ F'(x) = f(x)$), then: $\\ \int_a^b f(x) \, dx = F(b) - F(a)$

Interpretation: This part of the theorem allows us to evaluate the definite integral of $\ f(x)$ over $[a, b]$ by finding any antiderivative $\ F(x) \ of \ f(x)$, and then computing the difference $\ F(b) - F(a) .$

Example 1: Applying FTC1

Let $\ f(x) = 2x ,$ and define $\ F(x)$ as: $F(x) = \int_0^x 2t \, dt$

• To find $\ F(x)$, we integrate: $F(x) = \left. t^2 \right|_0^x = x^2$
• Now, differentiate $\ F(x)$ to verify FTC1: $F'(x) = \frac{d}{dx}(x^2) = 2x$
• As expected, $\ F'(x) = f(x)$, confirming that the derivative of the integral gives back the original function.

Example 2: Applying FTC2

Evaluate the definite integral: $\int_1^4 3x^2 \, dx$

• First, find an antiderivative $\ F(x) \ of \ f(x) = 3x^2$: $F(x) = x^3$
• Now, apply FTC2: $\int_1^4 3x^2 \, dx = F(4) - F(1) = 4^3 - 1^3 = 64 - 1 = 63$
• So, the value of the definite integral is 63.

• The Fundamental Theorem of Calculus connects differentiation and integration, showing that they are essentially inverse processes.
• FTC1 tells us that the derivative of the integral of a function is the function itself.
• FTC2 allows us to evaluate definite integrals by finding an antiderivative and computing the difference in its values at the endpoints.
• This theorem is central to the practice of calculus and has profound implications in many areas of mathematics, physics, engineering, and beyond.