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Integrals and Area
Integrals and Area
Integration is the reverse of differentiation and is also used to measure accumulation and signed area.
Indefinite Integral
$$\int f(x)\,dx = F(x)+C$$ where $F'(x)=f(x)$.
Basic Antiderivatives
| Function | Integral |
|---|---|
| $x^n$ ($n \ne -1$) | $\frac{x^{n+1}}{n+1}+C$ |
| $e^x$ | $e^x+C$ |
| $\cos x$ | $\sin x+C$ |
| $\sin x$ | $-\cos x+C$ |
Example 1
Evaluate $$\int (3x^2-4)\,dx.$$
$$x^3-4x+C$$
Definite Integral
$$\int_a^b f(x)\,dx$$ represents signed area from $x=a$ to $x=b$.
Fundamental Theorem of Calculus
If $F'(x)=f(x)$, then $$\int_a^b f(x)\,dx = F(b)-F(a).$$
Example 2
Evaluate $$\int_0^2 2x\,dx.$$
An antiderivative is $x^2$. Then $$x^2\big|_0^2=4-0=4.$$
Practice Problems
1. $$\int x^4\,dx$$
2. $$\int (5x-2)\,dx$$
3. $$\int e^x\,dx$$
4. $$\int_1^3 x\,dx$$
5. What does the constant $C$ represent?
Show Answer Key
1. $\frac{x^5}{5}+C$
2. $\frac{5}{2}x^2-2x+C$
3. $e^x+C$
4. $4$
5. The family of all antiderivatives differs by a constant.