5 Integrals and Their Applications Free Online FlashCards

What is an antiderivative of $f$?

An antiderivative of $f$ is a function $F$ whose derivative satisfies $F'(x)=f(x)$.

Why does an indefinite integral include $+C$?

The constant $C$ is required because every constant differentiates to zero; therefore all antiderivatives differ by an arbitrary constant.

State the power rule for integration.

For $n\ne-1$, $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$.

What does a definite integral represent?

A definite integral is a number representing signed area or accumulated net change. Regions below the $x$-axis contribute negatively.

What is the form of a Riemann sum?

A Riemann sum is $\sum_{i=1}^{n}f(x_i^*)\Delta x$, where $x_i^*$ is a sample point in subinterval $i$.

How is the width of equal Riemann-sum subintervals computed?

For $n$ equal subintervals of $[a,b]$, the width is $\Delta x=\frac{b-a}{n}$.

What does the Fundamental Theorem Part 1 state?

If $f$ is continuous, then $\frac{d}{dx}\int_a^x f(t)\,dt=f(x)$.

What does the Fundamental Theorem Part 2 state?

If $F'(x)=f(x)$ on $[a,b]$, then $\int_a^b f(x)\,dx=F(b)-F(a)$.

How do you differentiate an integral with upper limit $g(x)$?

For continuous $f$, $\frac{d}{dx}\int_a^{g(x)}f(t)\,dt=f(g(x))g'(x)$.

What happens when the limits of integration are reversed?

Reversing the limits changes the sign: $\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$.

How do definite integrals combine over adjacent intervals?

Adjacent intervals combine by additivity: $\int_a^b f(x)\,dx+\int_b^c f(x)\,dx=\int_a^c f(x)\,dx$.

How can bounds on $f$ bound its definite integral?

If $m\le f(x)\le M$ on $[a,b]$, then $m(b-a)\le\int_a^b f(x)\,dx\le M(b-a)$.