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5.5 Application of Integration

Tags
Integration
Area
Inverse function
Area between two functions
Solids of revolution
Kinematics
Trigonometric functions
Exponential and Logarithmic functions
Intercepts
By parts
Substitution
Partial fractions
Related rates
Complex numbers
Differentiation
Chain rule
Product rule
Quotient rule
There are 3 different applications you need to be aware of.
Application
Steps
Area
Area below the curve is - whereas the area above the curve is ++.
The area between two functions: ∫ab[f(x)−g(x)]dx\int _a^b[f(x) -g(x)] dx given f(x)>g(x)f(x) > g(x).
For example, the total area of this would be:
∫−30f(x)dx−∫01f(x)dx=∫−31∣f(x)∣dx\int _{-3}^0f(x) dx -\int _0^1f(x) dx=\int _{-3}^1|f(x)| dx.
Similarly, the area between the curve and the yy-axis is: ∫cdf−1(y)dy\int_c^df^{-1}(y) dy.
Area right to the curve is −- whereas the area left to the curve is ++.
Change in quantity
∫abf(x)dx\int _a^bf(x) dx denotes the overall change in quantity over the interval [a,b][a, b].
Area between a curve and the yy-axis (HL)
The area between a curve and the y-axis on the interval cyd can be found by using the formula:
A=∫cdf−1(y)dyA=\int _c^df^{-1}(y) dy
y=f(x)y=f(x) must be rearranged into x=g(y)=f−1(y)x=g(y)=f^{-1}(y), similar to finding the inverse function.
Solids of revolutions (HL)
A solid of revolution is formed when an area bounded by y=f(x)y=f(x) is rotated about the xx-axis through 2 radians, creating a 3-dimensional solid.
Volume of solid revolved by the xx-axis for a≤x≤ba\leq x\leq b:
V=π∫ab[f(x)]2dxV=π\int_a^b[f(x)]^2 dx
Similarly, if the function is invertible, an area bounded by y=f(x)y=f(x) can be rotated about the yy-axis through 2 radians form a solid of revolution.
Volume of solid revolved by the yy-axis for a≤y≤ba\leq y\leq b:
V=π∫ab[f−1(y)]2dyV=π\int_a^b[f^{-1}(y)]^2 dy
Proof:
A solid of revolution can be considered as being made up of infinite number of thin cylindrical discs, where the radius of each disc is x=xix=x_i and the height is hh. Since the volume of ii th disc is π[f(xi)]2h\pi [f(x_i)]^2h, the total volume of the disc is ∑i[f(xi)]2h\sum_i[f(x_i)]^2h.
Hence, the exact volume of the solid of revolution can be defined as:
V=lim⁡h→0∑i[f(xi)]2hV=\lim\limits_{h\rightarrow 0}\sum_i[f(x_i)]^2h
=∫ab[f(x)]2=π∫ab[f(x)]2dx=\int _a^b[f(x)]^2=\pi \int_a^b[f(x)]^2dx
Volumes for two defining functions
Volume of solid revolved by the xx-axis by 2 functions: π∫ab[f(x)2−g(x)2]dxπ\int_a^b[f(x)^2-g(x)^2] dx given f(x)>g(x)f(x) > g(x) in that interval. (similar to the area)
5.6 Applications of integration
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5.5 Application of Integration