# Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 4.9 – Antiderivatives

• Section 5.3 – The Fundamental Theorem of Calculus

• Section 5.4 – Indefinite Integrals and the Net Change Theorem

### Expected Educational Results

• Objective 01–01: I can state the Fundamental Theorem of Calculus.

• Objective 01–02: I can explain the meaning of the Fundamental Theorem of CalculusPart I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of CalculusPart II.

• Objective 01–04: I can determine the properties of an area function using FTC-I and Calculus I.

• Objective 01–05: I can evaluate indefinite integrals using the Fundamental Theorem of CalculusPart I.

• Objective 01–06: I can evaluate definite integrals using the Fundamental Theorem of CalculusPart II.

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_{a}^{b}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)\phantom{\rule{0.167em}{0ex}}\text{d}x$$\int_a^b{f^{\,\prime}(x)\,\text{d}x}$.

• Objective 01–08: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.

• Objective 01–09: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

### Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

## FTC-I

Recall, from CPT_01c_FTC_I.html,

#### Theorem: Fundamental Theorem of Calculus - Part I

Fundamental Theorem of Calculus - Part I: Let $f\left(t\right)$$f(t)$ be continuous on $\left[a,b\right]$$[a,b]$ and for any $x$$x$ in $\left[a,b\right]$$[a,b]$, let $F\left(x\right)={\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle F(x)=\int_a^x{f(t)\,dt}$. Then $F\left(x\right)$$F(x)$ is differentiable on $\left[a,b\right]$$[a,b]$ and ${F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=f\left(x\right)$$F^{\,\prime}(x)=f(x)$, for any $x$$x$ in $\left[a,b\right]$$[a,b]$.

Note: The variable $x$$x$ is the independent variable, and the variable $t$$t$ is called a dummy variable.

Conditions to be met:

1. $f\left(t\right)$$f(t)$ is continuous for all values on $\left[a,b\right]$$[a,b]$;

2. $F\left(x\right)$$\displaystyle F(x)$ is defined as the integral ${\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle \int_a^x{f(t)\,dt}$, where

• a constant, $a$$a$, is the lower limit;

• the variable, $x$$x$, and not some function of $x$$x$, is the upper limit;

• the integral is in terms of a dummy variable, e.g., $t$$t$, and not $x$$x$.

Conclusion of FTC-I:

If the above conditions are met, then the conclusion of the FTC-I is

${F}^{\phantom{\rule{0.222em}{0ex}}\prime }\left(x\right)=\frac{d}{dx}{\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.222em}{0ex}}dt=f\left(x\right)$$\displaystyle F^{\>\prime}(x)=\frac{d}{dx}\int_a^x{f(t)\>dt}=f(x)$.

#### Activity 11

Use what you learned from the previous Activity page (ACT_01e_Derivative_of_Integral_Function.html). Without using any technology, differentiate the following:

1. $g\left(x\right)={\int }_{2}^{x}\sqrt{{t}^{2}+t-1}\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle g(x) = \int_2^x{\sqrt{t^2 + t - 1}\, dt}$. Explain. Check your answer with the embedded quiz below.

2. $h\left(x\right)={\int }_{x}^{1}\frac{t+1}{{t}^{2}+1}\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle h(x) = \int_x^1{\frac{t+1}{t^2+1}\, dt}$. Explain. Check your answer with the embedded quiz below.

3. $k\left(x\right)={\int }_{1}^{{x}^{2}}\mathrm{tan}\left(t+1\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle k(x) = \int_1^{x^2}{\tan{(t+1)}\, dt}$. Explain. Check your answer with the embedded quiz below.

4. $m\left(x\right)={\int }_{1/x}^{-1}\mathrm{cos}\left(2-t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle m(x) = \int_{1/x}^{-1}{\cos{(2-t)}\, dt}$. Explain.

## Homework

At this time, you should be able to complete the following assignments:

• Section 5.3: # 7, 9, 11, 13, 15, 17, 19.