Find the shaded region in the graph. The shaded area is the difference between the area of the and rectangle bounded by the x- and y-axes, y=3 and x= 4 the area bounded by the x-axis, the y-axis, y=2 cos x+1, 4 4- and x= 4 y=3 noo ai olao to meO9 3n The area of the rectangle is 2- ei oviteo 4 s o) no slonnenb ons y=2 cos x+1 1- x/4 (x v The area under the curve is (2 cos x+1)dx. rT 1/2 Evaluate this integral. 1/4 v of msot iel smon it giviunA Please explain each Step in detail. thank you (2 cos x+1)dx [2 sin x+: a/4 = 2 + Thus, the shaded area is 2 V2.
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.
![The image contains a graph and a mathematical explanation regarding finding the shaded area in a graph. Here is a transcription suitable for an educational website:
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**Find the Shaded Region in the Graph**
The problem involves finding the shaded area, which is described as the difference between two specific areas:
1. **Rectangle Bounded**:
- Bounded by the x-axis, y-axis, y = 3, and x = π/4.
- The area of this rectangle is calculated as:
\[
\text{Area of Rectangle} = 3 \cdot \frac{\pi}{4} = \frac{3\pi}{4}
\]
2. **Area Bounded by a Curve**:
- Bounded by the x-axis, the y-axis, and the curve y = 2cos(x) + 1.
- Evaluate the integral to find this area:
\[
\int_{0}^{\pi/4} (2\cos x + 1) \, dx
\]
**Steps to Solve**:
- Evaluate the integral:
\[
\int_{0}^{\pi/4} (2\cos x + 1) \, dx = [2\sin x + x]_{0}^{\pi/4} = (\sqrt{2} + \frac{\pi}{4})
\]
- Determine the shaded area by subtracting the area under the curve from the area of the rectangle:
\[
\text{Shaded Area} = \frac{3\pi}{4} - \left( \sqrt{2} + \frac{\pi}{4} \right) = \frac{\pi}{2} - \sqrt{2}
\]
**Explanation of the Graph**:
- The graph displays two functions and various markings:
- The horizontal and vertical boundaries are highlighted to represent the bounds for the area calculations.
- The function y = 2cos(x) + 1 creates a curve intersecting with y = 3 at specific x-values.
The handwritten note on the graph reads: "Please explain each step in detail. Thank you!"
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This guide helps enumerate each critical step and decision that leads to determining the shaded area, supporting learners in understanding the fundamental concepts of area integration and geometry within the context of graph and function analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeadb183-1175-48ee-be1d-be4cbd36e1e9%2F0030bb8a-9f29-4447-8485-ce7c8e876eb3%2F5qv3htd_processed.jpeg&w=3840&q=75)
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