Calculus Notes

Definite Integral

• The definite integral is represented as $\int_{a}^{b} f(x) \, dx$, where:
  • $a$ and $b$ are the limits of integration, representing the starting and stopping points of $x$.
  • $f(x)$ is the function to be integrated.
• This notation represents the area under the curve of $f(x)$ from $x = a$ to $x = b$.
• Instead of calculating the area through an infinite sum, we find an antiderivative $F(x)$ of $f(x)$.
• The fundamental theorem of calculus states that $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$ (i.e., $F'(x) = f(x)$).

Example

• Evaluate the definite integral $\int_{-1}^{2} (4 - x^2) \, dx$.
• First, find the antiderivative $F(x)$ of $f(x) = 4 - x^2$.
  • The antiderivative of 4 is $4x$.
  • The antiderivative of $-x^2$ is $-\frac{1}{3}x^3$.
  • Therefore, $F(x) = 4x - \frac{1}{3}x^3$.
• Now, evaluate $F(2) - F(-1)$.
  • $F(2) = 4(2) - \frac{1}{3}(2)^3 = 8 - \frac{8}{3} = \frac{24}{3} - \frac{8}{3} = \frac{16}{3}$.
  • $F(-1) = 4(-1) - \frac{1}{3}(-1)^3 = -4 + \frac{1}{3} = -\frac{12}{3} + \frac{1}{3} = -\frac{11}{3}$.
  • $\int_{-1}^{2} (4 - x^2) \, dx = F(2) - F(-1) = \frac{16}{3} - (-\frac{11}{3}) = \frac{16}{3} + \frac{11}{3} = \frac{27}{3} = 9$.

Indefinite Integral

• The indefinite integral is written as $\int f(x) \, dx$.
• It represents the family of all antiderivatives of $f(x)$.
• If $F(x)$ is an antiderivative of $f(x)$, then the indefinite integral is given by $\int f(x) \, dx = F(x) + C$, where C is an arbitrary constant.
• The constant C accounts for the fact that the derivative of a constant is zero, so any constant can be added to an antiderivative and it will still be a valid antiderivative.
• Example: If $F(x) = x^2$ is an antiderivative of $f(x) = 2x$, then $x^2 + \pi$, $x^2 - 1000$, and $x^2$ plus any other real number you can think of are valid anti-derivatives.

Signed Area

• The definite integral calculates the signed area between a curve and the x-axis.
• Areas above the x-axis are considered positive, while areas below the x-axis are considered negative.

Example: Sine Function

• Area under the first hump of the sine function:
  • $\int<em>{0}^{\pi} \sin(x) \, dx = [-\cos(x)]</em>{0}^{\pi} = -\cos(\pi) - [-\cos(0)] = -(-1) - (-1) = 1 + 1 = 2$
  • Here, the area is positive because the region is above the x-axis.

Direction of Integration

• The direction of integration matters for signed areas.

• If b > a, then $\Delta x = \frac{b - a}{n}$ is positive.

• If a > b, then $\Delta x = \frac{b - a}{n}$ is negative, which results in a negative area if the function is positive over the interval and vice versa.

Properties of Definite Integrals

• Integral from a point to itself:
  • $\int_{a}^{a} f(x) \, dx = 0$
• Integral over adjacent intervals:
  • $\int<em>{a}^{c} f(x) \, dx + \int</em>{c}^{b} f(x) \, dx = \int_{a}^{b} f(x) \, dx$
  • This property holds regardless of where $c$ is located (i.e., whether $c$ is between $a$ and $b$ or outside the interval).