Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = e-5x y = 0 x = 0 X = 3
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 volume of the solid generated by revolving the region bounded by the given graphs about the x-axis, follow these steps.
The equations defining the region are:
\[ y = e^{-5x} \]
\[ y = 0 \]
\[ x = 0 \]
\[ x = 3 \]
**Explanation:**
We need to find the volume of the solid formed when this region is revolved about the x-axis. This type of volume calculation can be done using the method of disks or washers.
**Steps:**
1. **Identify the function and bounds:**
The function \( y = e^{-5x} \) is bounded by \( x = 0 \) and \( x = 3 \), and the line \( y = 0 \) forms the lower boundary.
2. **Setup the Disk Method Integral:**
When revolving around the x-axis, the radius of a cylindrical disk at a point \( x \) is given by the function \( y = e^{-5x} \). The volume of each disk is \( \pi \cdot [radius]^2 \cdot [thickness] \). Hence, we need to integrate the area of these disks from \( x = 0 \) to \( x = 3 \).
3. **Write the Volume Integral:**
\[
V = \pi \int_{0}^{3} (e^{-5x})^2 \, dx
\]
4. **Simplify the Integrand:**
Simplify the expression inside the integral:
\[
V = \pi \int_{0}^{3} e^{-10x} \, dx
\]
5. **Evaluate the Integral:**
To integrate \( e^{-10x} \), we use the formula for the integral of the exponential function:
\[
\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C
\]
Substituting \( a = -10 \):
\[
\int e^{-10x} \, dx = -\frac{1}{10} e^{-10x} + C
\]
6. **Apply the Bounds:**
Evaluate the definite integral from \( 0 \) to \( 3 \):
\[
V = \pi \left[ -\frac{1}{10} e](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad118f04-c8c0-46fe-8652-72d195094665%2F79a9f2e9-75af-4052-9db9-720e19651a13%2Fu2ngm5d_processed.jpeg&w=3840&q=75)
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