Learning Outcomes

  1. Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals. 
  2. Use the Fundamental Theorem of Calculus, Part 2,  to evaluate definite integrals. 
  3. Explain the relationship between differentiation and integration. 
     

Textbook

  • Chapter 5.3 ¬†The Fundamental Theorem of Calculus ¬†

Textbook Assignment

  • p. 562: ¬† ¬†170, 171, 177, 182, 183¬†

WeBWork Assignment

  • Integration-Fundamental Theorem

Exit problems of the session 

  1. Find the derivative  using the Fundamental Theorem of Calculus:

    (a).  \dfrac{d}{dx}\displaystyle\int_1^x 3te^{t^2}dt    (b).   \dfrac{d}{dx}\displaystyle\int_0^x \sin^2 t   dt  
  2. Evaluate following definite integrals. 
    (a).  \displaystyle\int_1^3 3x(x^2-1)dx    (b).   \displaystyle\int_1^4 \dfrac{x^2+\sqrt{x}}{x}dx     (c).   \displaystyle\int_0^1 (3e^t-2\cos t)\;dt

 

Key Concepts

  • Fundamental Theorem of Calculus, Part 1:¬†

If f(x) is a continuous function over an interval  [a, b],  and the function F(x) is defined by 

F(x)=\displaystyle\int_a^x f(x) dx,    then     F'(x)=f(x).   

  • Fundamental Theorem of Calculus, Part 2:

If f(x) is a continuous function over an interval  [a, b],   and F(x) is any antiderivative of f(x),  then 

\displaystyle\int_a^b f(x) dx=F(b)-F(a).

 

Videos and Practice Problems of Selected Topics

  1. \rhd The Fundamental Theorem of Calculus (8:02) Connecting differentiation and integration.
  2. \rhd Finding the derivative using the Fundamental Theorem of Calculus (3:23) Find \dfrac{d}{dx}\displaystyle\int_{19}^x\sqrt[3]t dt.
  3. * Practice: Finding the derivative using the Fundamental Theorem of Calculus. (4 problems)
  4. \rhd Definite integrals: reverse power rule (4:13) Find \displaystyle\int_{-3}^{5}4dx  and \displaystyle\int_{-1}^{3}7x^2dx.
  5. * Practice:  Definite integrals: reverse power rule. (4 problems)
  6. \rhd Definite integrals: rational functions (5:04) Find \displaystyle\int_{-1}^{2}\dfrac{16-x^3}{x^3}dx.
  7. \rhd Definite integrals: radical functions (3:50) Find \displaystyle\int_{-1}^{8}12\sqrt[3]xdx.
  8. \rhd Definite integrals: trigonometric functions (4:59)  Find \displaystyle\int_{11\pi/2}^{6\pi}9\sin(x)dx.
  9. \rhd Definite integrals: logarithmic functions (7:26)  Find \displaystyle\int_{2}^{4}\dfrac{6+x^2}{x^3}dx.
  10. *Practice: Definite integrals. (4 problems)