Integrals are mathematical tools that allow us to calculate the total accumulation of a quantity over a range or find the area under a curve. They do this by dividing the quantity into infinitesimally small parts and adding them together. Specifically, integrals are used to find the area under a curve, relate velocity to displacement, calculate volumes of 3D objects, find critical points in optimization problems, calculate probabilities and expected values in statistics, and have many real-world applications like determining fluid flow and signal processing.

1. Applications of
Integrals
BY AADHITHYA S ESWAR
MCT A
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2. What are integrals
 Integrals are mathematical tools used to calculate the total accumulation of a
quantity over a range or to find the area under a curve.
 In simple terms, they help us determine the sum or accumulation of
infinitesimally small parts.
 Integrals are denoted by the symbol ∫ and have a function to be integrated,
as well as limits that define the range of integration.
 By evaluating an integral, we can find the value of the accumulated quantity,
or the area enclosed by a curve.
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3. Area under a Curve
 Dividing into infinitesimally small parts:
Integrals allow us to divide the area under a curve
into an infinite number of tiny, vertical strips. By
summing up the areas of these strips using
integration, we can calculate the total area under
the curve.
 Accumulating the contributions: Each
infinitesimally small strip contributes a small area
to the total, and integration helps us accumulate
these contributions to find the overall area.
Integrals essentially add up all the tiny areas to
give us the final value representing the area under
the curve.

4. Displacement and
Distance
 Connecting velocity and displacement: Integrals can relate
the velocity of an object to its displacement. By integrating the
velocity function with respect to time, we obtain the
displacement function. This integration process accounts for
the changes in velocity over time and yields the net change in
position, giving us the object's displacement.
 Absolute value for distance: To find the total distance
traveled by an object, integrals can be used. By taking the
integral of the absolute value of the velocity function over a
specific time interval, we accumulate the magnitudes of the
velocities, disregarding their directions. This approach allows
us to account for both positive and negative velocities and
provides the total distance traveled by the object.

5. Volume of 3D Objects
 Slicing the object: Integrals help us calculate the
volume of a 3D object by slicing it into infinitesimally
thin, parallel cross-sectional slices. Each slice can be
approximated as a 2D shape, such as a circle, square,
or any other shape depending on the object's cross-
section.
 Summing the slices: By integrating the areas of
these infinitesimally thin slices over the entire object,
we can sum up the volumes of each slice. The
integral adds together the contributions of all the
slices, allowing us to obtain the total volume of the
3D object.

6. Optimization with
Integrals
 Finding critical points: Integrals play a crucial role in optimization by
helping us find critical points, which are points where a function reaches
a maximum or minimum. By taking the derivative of a function and
setting it equal to zero, we can determine critical points. Integrals come
into play by allowing us to analyze the behavior of the function in the
neighborhood of these critical points and identify whether they
correspond to maximum or minimum values.
 Evaluating objective functions: Integrals assist in optimization by
enabling us to evaluate objective functions. Objective functions
represent the quantity we aim to maximize or minimize. By integrating
the objective function over a specified domain or range, we obtain a
single value that quantifies the total or average value of the function.
This allows us to compare different solutions or scenarios and select the
optimal one based on the desired criteria.

7. Probability and Statistics
 Probability density functions (PDFs): Integrals are used to calculate
probabilities in continuous probability distributions. Probability density
functions (PDFs) represent the distribution of a continuous random
variable. The area under the PDF curve between two points represents
the probability of the random variable falling within that range. By
integrating the PDF over a specific interval, we can determine the
probability of an event occurring within that interval.
 Expected values: Integrals play a key role in calculating expected
values in probability and statistics. The expected value represents the
average value of a random variable. By multiplying each possible
outcome of a random variable by its corresponding probability and
summing them up, we can calculate the expected value. For continuous
random variables, the summation is replaced by an integral, allowing us
to determine the average value over a continuous range.

8. Real-World Applications
 Area of irregular shapes: Integrals are used to calculate the area of irregular shapes or regions. For example, in land
surveying, integrals can be used to determine the area of a non-rectangular plot of land.
 Fluid flow and volume: Integrals are employed in fluid mechanics to calculate fluid flow rates and determine the
volume of fluid passing through a given area or pipe. This is vital in engineering fields such as civil engineering,
where water flow in pipes or channels needs to be accurately estimated.
 Population dynamics: Integrals are used to model and analyze population dynamics in biology and ecology. By
integrating growth and mortality rates, researchers can estimate population sizes and predict future trends.
 Electrical circuits: Integrals play a role in analyzing and designing electrical circuits. They are used to calculate
quantities such as voltage, current, and power, helping engineers optimize circuit performance and ensure proper
functioning.
 Signal processing: Integrals are used in signal processing to analyze and manipulate signals. For example, in audio
and image processing, integrals are employed to calculate the energy or power of a signal, filter out unwanted
noise, and extract relevant information.

9. Thank you