To find the area of the shaded region, we need to compute the area for region A on the interval [0, 2] and the area for region B on the interval [2, 4], and then find the sum of the regions. y y =x - 2 x + 1 2 4 y=5 - x For region A, the interval from [0, 2], notice that y = 5- equations and limits of integration can be substituted into the area formula. x is above the graph of y = x² – 2x + 1, so the Area = [f(x) – g(x)] dx ((5 – x2) – (x² – 2x + 1)] dx Area of A = substitute -2x² + 2x + dx %3D distribute and combine like terms +x² + ax %3D integrate - - + (2)* + 4(2) - -0 + (o)? + 40) evaluate 3. simplify Thus, the area of region A is square units.
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![To find the area of the shaded region, we need to compute the area for region A on the interval [0, 2] and the
area for region B on the interval [2, 4], and then find the sum of the regions.
y
y =x - 2 x + 1
y=5- x
For region A, the interval from [0, 2], notice that y = 5 - x is above the graph of y = x - 2x + 1, so the
equations and limits of integration can be substituted into the area formula.
Area =
[f(x) – g(x)] dx
[(5 – x3) – (x² – 2x + 1)] dx
Area of A =
substitute
-2x² + 2x +
dx
distribute and combine like terms
+ x2
+ 4x
integrate
- -2)? + (2)? + 4(2) - -0 - (o)? + 4co)
%3D
evaluate
simplify
Thus, the area of region A is
square units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b29fba6-eb9e-418e-8a6d-3991cb107cb2%2Fe8521186-0966-4cda-8d05-a1a2879ffa8f%2F7i8v67l_processed.jpeg&w=3840&q=75)
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