# Calculus UNIT508: What is the Fundamental Theorem of Calculus?

## 2943 days ago by MATH4R2013

#1) (FTC I) why does integrate(5,x,1,4) = 15? show(integrate(5,x,1,4)) plot(5,1,4)
 \newcommand{\Bold}{\mathbf{#1}}15  #2) (FTC I) why does integrate(sqrt(4-x^2),x,-2,2) = 2*pi? show(integrate(sqrt(4-x^2),x,-2,2)) plot(sqrt(4-x^2),-2,2)
 \newcommand{\Bold}{\mathbf{#1}}2 \, \pi  #3) (FTC II) given f(x)=integrate(t^2,t,0,x), what is f'(x)? var('t') f(x)=integrate(t^2,t,0,x) g(x)=diff(f(x),x) show(f(x)) show(g(x))
 \newcommand{\Bold}{\mathbf{#1}}\frac{1}{3} \, x^{3} \newcommand{\Bold}{\mathbf{#1}}x^{2}
#4) (FTC II) given f(x)=integrate(cos(t),t,-pi,x), what is f'(x)? f(x)=integrate(cos(t),t,-pi,x) g(x)=diff(f(x),x) show(f(x)) show(g(x))
 \newcommand{\Bold}{\mathbf{#1}}\sin\left(x\right) \newcommand{\Bold}{\mathbf{#1}}\cos\left(x\right)
#5) (FTC II and Chain Rule) given f(x)=integrate(cos(t),t,1,x^2), what is f'(x)? f(x)=integrate(cos(t),t,-pi,x^2) g(x)=diff(f(x),x) show(f(x)) show(g(x))
 \newcommand{\Bold}{\mathbf{#1}}\sin\left(x^{2}\right) \newcommand{\Bold}{\mathbf{#1}}2 \, x \cos\left(x^{2}\right)