Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. - 3 sin(), y - Şx y - y y 4 2 2 -4 -2 2 -2 -4 -6F y 6 4 3 2 -4 -5- 3 Find the area of the region.
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.
![**Title: Calculating the Area Bounded by Given Curves**
---
**Problem Statement:**
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) or \( y \). Draw a typical approximating rectangle. The equations of the curves are given by:
\[ y = 3 \sin \left(\frac{\pi x}{7}\right) \]
\[ y = \frac{6}{7} x \]
**Visual Representation of the Problem:**
The figures below illustrate the region enclosed by the two curves. Each figure also includes a typical approximating rectangle used for integration.
**Figure Descriptions:**
- **Top Left Graph:**
- The region enclosed by the curves \( y = 3 \sin \left(\frac{\pi x}{7}\right) \) and \( y = \frac{6}{7} x \) is shaded in blue.
- The graph shows the curves extending symmetrically about the origin, with intersections marked.
- A red rectangle is drawn vertically, indicating integration with respect to \( x \).
- **Top Right Graph:**
- Another representation of the enclosed region, focusing on the area near the upper intersection.
- The region remains shaded in blue, and the red rectangle suggests integration with respect to \( y \).
- **Bottom Left Graph:**
- The enclosed region is drawn near the lower intersection of the curves.
- A vertical red rectangle implies that integration with respect to \( y \) might be more suitable.
- **Bottom Right Graph:**
- This graph focuses again on the intersection near the positive y-axis.
- The blue shaded region is bounded by the same curves with the red rectangle drawn vertically for integration with respect to \( x \).
**Area Calculation:**
To calculate the area of the region:
1. **Find the intersection points** of the given curves by solving \( 3 \sin \left(\frac{\pi x}{7}\right) = \frac{6}{7} x \).
2. **Set up the definite integral** for the bounded region, using the appropriate limits determined by the points of intersection.
Depending on whether integration is more suitable with respect to \( x \) or \( y \):
- If integrating with respect to \( x \):
\[
\text{Area} = \int_{a}^{b} \left[ 3 \sin](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ecf1484-f95f-4119-9858-43c8382b92c0%2F6247f11f-72d3-4694-9d29-c30ac1e0fff6%2F6e252d8_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 3 images
