Mathematics

Definite Integral. Applications

Calculation of the definite integral and its applications: fundamental theorem of calculus, calculation of areas of planar regions, and volumes of revolution.

Definite integral: Given a function f(x)f(x) and an interval [a,b][a, b], the definite integral is equal to the area bounded by the graph of f(x)f(x), the x-axis, and the vertical lines x=ax=a and x=bx=b.
A=∫abf(x)dxA = \int_a^b f(x) dx
Area under a curve
Fundamental Theorem of Calculus (Barrow's Rule): The definite integral of a continuous function f(x)f(x) on a closed interval [a,b][a, b] is equal to the difference between the values of an antiderivative F(x)F(x) at the endpoints of the interval.
∫abf(x)dx=F(b)−F(a)\int_a^b f(x) dx = F(b) - F(a)
Properties of the definite integral
  • The value of the definite integral changes sign if the limits of integration are swapped.
    ∫abf(x)dx=−∫baf(x)dx\displaystyle \int_a^b f(x) dx = -\int_b^a f(x) dx
  • When the limits of integration are the same, the integral is zero.
    ∫aaf(x)dx=0\displaystyle \int_a^a f(x) dx = 0
  • If cc is an interior point of the interval [a,b][a, b], the integral can be split into the sum of two integrals over the intervals [a,c][a, c] and [c,b][c, b].
    ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx\displaystyle \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx
Mean Value Theorem for Definite Integrals
If a function f(x)f(x) is continuous on an interval [a,b][a, b], then there exists a point c∈[a,b]c \in [a, b] such that:
∫abf(x)dx=f(c)⋅(b−a)\int_a^b f(x) dx = f(c) \cdot (b-a)
Mean Value Theorem for Definite Integrals
Case 1. Area under a curve
A=∣∫abf(x)dx∣A = \left| \int_a^b f(x) dx \right|
Case 1. Area under a curve
Case 2. Area under a curve
The curve intersects the x-axis at c.
A=∣∫acf(x)dx∣+∣∫cbf(x)dx∣A = \left| \int_a^c f(x) dx \right| + \left| \int_c^b f(x) dx \right|
Case 2. Area under a curve with intersection on x-axis
Case 3. Area between 2 curves
A=∣∫ab[f(x)−g(x)]dx∣A = \left| \int_a^b [f(x) - g(x)] dx \right|
Case 3. Area between 2 curves
Case 4. Area between 2 intersecting curves
A=∣∫ab(f(x)−g(x))dx∣+∣∫bc(f(x)−g(x))dx∣A = \left| \int_a^b (f(x) - g(x)) dx \right| + \left| \int_b^c (f(x) - g(x)) dx \right|
Case 4. Area between 2 intersecting curves
Volumes of revolution:
A volume of revolution is obtained by rotating an arc of the function f(x)f(x) around the x-axis. The volume of the generated shape is calculated as:
V=π∫ab[f(x)]2dxV = \pi \int_a^b [f(x)]^2 dx
Volume of revolution
Calculation of arc length
L=∫ab1+[f′(x)]2dxL = \int_a^b \sqrt{1+[f'(x)]^2} dx
Area of a surface of revolution
A=∫ab2πf(x)1+[f′(x)]2dxA = \int_a^b 2\pi f(x) \sqrt{1+[f'(x)]^2} dx
Work done by a force
W=∫abF(x)dxW = \int_a^b F(x) dx