Lesson 18 The Fundamental Theorem of Calculus

18.1 Lecture Content

Video: The Fundamental Theorem of Calculus

18.2 Lecture Notes

18.2.1 Fundamental Theorem of Calculus – Part I

Let \(f\) be a continuous function on the interval \([a, b]\), and define a function:

\[ F(x) = \int_a^x f(t)\,dt \]

Then \(F\) is continuous on \([a, b]\), differentiable on \((a, b)\), and:

\[ F'(x) = f(x) \]

Interpretation:

The derivative of the area function \(F(x)\) gives back the original function \(f(x)\). This connects differentiation and integration.

18.2.2 Fundamental Theorem of Calculus – Part II

Let \(f\) be continuous on \([a, b]\), and let \(F\) be any antiderivative of \(f\) on that interval. Then:

\[ \int_a^b f(x)\,dx = F(b) - F(a) \]

Interpretation:

To evaluate a definite integral, find any antiderivative \(F\), and compute the difference \(F(b) - F(a)\). This allows us to compute exact areas without using limits or Riemann sums.

18.2.3 Properties of Definite Integrals

Let \(f(x)\), \(g(x)\) be continuous on \([a, b]\), and let \(c\) be a constant. The following properties hold:

  1. Linearity:

\[ \int_a^b [f(x) \pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx \]

\[ \int_a^b c \cdot f(x)\,dx = c \cdot \int_a^b f(x)\,dx \]

  1. Additivity over Intervals:

\[ \int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx \]

  1. Reversing Limits:

\[ \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx \]

  1. Zero-Width Interval:

\[ \int_a^a f(x)\,dx = 0 \]

  1. Comparison Property (if \(f(x) \ge g(x)\)):

\[ \int_a^b f(x)\,dx \ge \int_a^b g(x)\,dx \]

  1. Absolute Value Inequality:

\[ \left| \int_a^b f(x)\,dx \right| \le \int_a^b |f(x)|\,dx \]

18.2.4 Fundamental Theorem of Calculus

Video: Solutions

  1. Evaluate the definite integral by interpreting it in terms of areas \(\int_2^7 (2x - 12) \, dx\).

  2. Use the limit definition of the integral to evaluate \(\int_2^4 (x^2 + 4x - 3) \, dx\).

  3. Evaluate the integral \(\int_2^3 (x + 5)(x - 3) \, dx\)

  4. If \(f(x) = \int_x^{19} t^6 \, dt,\) then find \(f'(x)\)

  5. If \(f(x) = \int_4^x t^3 \, dt,\) then find \(f'(x)\). Also evaluate \(f'(5)\).

  6. If \(f(x) = \int_{-2}^x^3 t^2 \, dt,\) then find \(f'(x)\).

  7. Use Part I of the Fundamental Theorem of Calculus to find the derivative of \(f(x) = \int_4^x \left( \frac{1}{4} t^2 - 1 \right)^8 \, dt\). Find \(f'(x)\).

18.3 Practice Problems

  1. Evaluate the following integrals:

    1. \(\displaystyle \int_{-3}^{3} \left(x + x^3\right) dx\)
    2. \(\displaystyle \int_{0}^{\pi} \left(\sin u + \cos u\right) du\)
    3. \(\displaystyle \int_{-1}^{1} \left(3x^4 - 2x^{-2} + x^{-5} + 4\right) dx\)
    4. \(\displaystyle \int_{2}^{3} \left(2xe^{x^2 + 4} + \dfrac{5}{2x+8} \right) dx\)
    5. \(\displaystyle \int_{0}^{1} \frac{e^x + e^{-x}}{2} dx\)

  2. Find the derivative of:

    1. \(g(x) = \displaystyle \int_{0}^{x} \left(u +u^3\right) du\)
    2. \(g(x) = \displaystyle \int_{-5}^{x^2} \sqrt{1 + u^3} du\)
    3. \(g(x) = \displaystyle \int_{\ln(x)}^{e^x} \left(u^6 - 1/u^2\right) du\)

  3. (Applied) The marginal revenue of the xth box of flash cards sold is \(100e^{-0.001x}\). Find the revenue from items 101 through 1,000.

  4. (Challenge) Differentiate
    \[\int_{g(x)}^{h(x)} f(t) dt.\]

  5. (Challenge) Show that
    \[2 \leq \int_{0}^2 \sqrt{1 + x^3} dx \leq 6.\]

18.4 Self Assessment

Time yourself and try to solve the following within 20 minutes:

  1. \(\displaystyle \int_{0}^{\pi/2} \left(1 + x + \cos x\right) dx\)
  2. \(\displaystyle \int_{-1}^{1} x(x^2+1)^{10} dx\)
  3. \(g(x) = \displaystyle \int_{x^3}^{\pi} e^{u^3 + 4u} du\)
  4. \(g(x) = \displaystyle \int_{x}^{\pi/2} (1 + u + \cos u) du\)
  5. Given \(C(x) = 246.76 + \int_{0}^x 5t dt\), find the fixed cost and marginal cost at \(x=10\).

18.5 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Apply the FTCI to integrals.
Apply the FTCII to integrals.
Solve applied problems using the FTC.