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:    149, 151, 153, 170, 171, 173, 174, 177, 179-183 all

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)