3 to x 6 using a Left Endpoint Approximate the area under the curve graphed below from x = approximation with 3 subdivisions. (You will need to approximate the function values using the graph.) 4 3 2 3 4 5 6 7 8
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.
![**Approximation Using Left Endpoint Rule**
**Task:**
Approximate the area under the given curve from \( x = 3 \) to \( x = 6 \) using a Left Endpoint approximation with 3 subdivisions.
(Note: You will need to estimate the function values using the graph provided.)
**Graph Description:**
- The graph displays a curve starting from the origin, curving upwards as \( x \) increases. The curve appears to be increasing at a decreasing rate.
- The x-axis ranges from -1 to 8, while the y-axis ranges from -1 to 5.
- The curve passes through points approximately at (1, 1), (2, 2), and (4, 3), with smooth curves connecting them.
**Approach for Left Endpoint Approximation:**
1. **Subdivisions:** Divide the interval [3, 6] into 3 equal parts:
- \([3, 4]\), \([4, 5]\), and \([5, 6]\).
2. **Function Values at Left Endpoints:**
- Approximate the y-values (heights of rectangles) from the graph at \( x = 3 \), \( x = 4 \), and \( x = 5 \).
3. **Area Calculation:**
- For each subdivision, multiply the height (approximated function value at the left endpoint) by the width (length of each subdivision, which is 1).
- Sum up the areas of the rectangles to get the total approximation.
Using this method, calculate the approximate area under the curve by visually inspecting the graph to determine the y-values at the left endpoints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb155383f-1482-4158-bfc0-94ad91579101%2Fbbc1de44-224e-4a26-8591-cec8b7229b89%2Fx80rbk_processed.png&w=3840&q=75)
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