6 The Fundamental Theorem of Calculus Free Online FlashCards

What is an antiderivative of $f$?

An antiderivative of f is a function F satisfying $F'(x)=f(x)$.

What does the constant $C$ represent in an indefinite integral?

An indefinite integral represents a family of antiderivatives and includes an arbitrary constant: $\int f(x)\,dx=F(x)+C$.

What does a definite integral measure?

A definite integral measures signed accumulation, or net area: contributions above the axis are positive and those below it are negative.

What does FTC Part 1 say about $G(x)=\int_a^x f(t)\,dt$?

If $G(x)=\int_a^x f(t)\,dt$ and f is continuous, then $G'(x)=f(x)$.

Why is $t$ called a dummy variable in $\int_a^x f(t)\,dt$?

The dummy variable t distinguishes the integration variable from the variable upper limit x; it can be replaced by another variable without changing the integral.

State FTC Part 2.

For continuous f with antiderivative F, $\int_a^b f(x)\,dx=F(b)-F(a)$.

Evaluate $\int_1^3(2x^3-4x+1)\,dx$.

An antiderivative is $F(x)=\frac{x^4}{2}-2x^2+x$, so $F(3)-F(1)=24$.

Should you include $+C$ when evaluating a definite integral?

No. In a definite integral, the constants cancel: $(F(b)+C)-(F(a)+C)=F(b)-F(a)$.

What does the Net Change Theorem state?

The net change equals the integral of the rate: $Q(b)-Q(a)=\int_a^b Q'(t)\,dt$.

Find the displacement when $v(t)=3t^2-4t$ from $t=1$ to $t=3$.

The displacement is $\int_1^3(3t^2-4t)\,dt=10\text{ m}$.

How does total distance differ from displacement?

Total distance uses speed: $\int_a^b|v(t)|\,dt$, or it splits the interval wherever velocity changes sign.

For $f(x)=x-1$ on $[0,2]$, what is the total geometric area?

The net area is $0$, but the total geometric area is $\int_0^1(1-x)\,dx+\int_1^2(x-1)\,dx=1$.