To solve this problem, let's break it down step-by-step:

1. Understanding the Curves:

• We are given two functions: $f(x) = \cosh(x) \quad \text{and} \quad g(x) = x$ as well as the vertical boundaries $x = 0$ and $x = 3$.

The region of interest is the area bounded by these curves between $x = 0$ and $x = 3$.

2. Sketching the Curves:

• The function $\cosh(x)$ (the hyperbolic cosine) is symmetric and grows exponentially for large values of $x$. It looks similar to an upward-opening parabola but grows faster.
• The function $g(x) = x$ is a straight line passing through the origin with a slope of 1.
• You are asked to find the area between these two curves from $x = 0$ to $x = 3$.

3. Setting Up the Integral:

The area between two curves is given by the formula: $\text{Area} = \int_{a}^{b} \left( f(x) - g(x) \right) \, dx$ Here, $f(x) = \cosh(x)$ and $g(x) = x$, and the bounds are $x = 0$ to $x = 3$. Therefore, the area is: $\text{Area} = \int_{0}^{3} \left( \cosh(x) - x \right) \, dx$

4. Computing the Integral:

We can now compute the integral: $\text{Area} = \int_{0}^{3} \cosh(x) \, dx - \int_{0}^{3} x \, dx$

The first integral is straightforward because the derivative of $\sinh(x)$ is $\cosh(x)$: $\int \cosh(x) \, dx = \sinh(x)$ So: $\int_{0}^{3} \cosh(x) \, dx = \sinh(3) - \sinh(0) = \sinh(3)$

The second integral is a basic polynomial integral: $\int x \, dx = \frac{x^2}{2}$ So: $\int_{0}^{3} x \, dx = \frac{3^2}{2} - \frac{0^2}{2} = \frac{9}{2}$

Putting these together: $\text{Area} = \sinh(3) - \frac{9}{2}$

Using the value $\sinh(3) \approx 10.0179$, we get: $\text{Area} \approx 10.0179 - 4.5 = 5.5179 \, \text{square units}$

5. Finding the Mass:

The mass density is given as 7 grams per unit area. Therefore, the mass of the part is: $\text{Mass} = \text{density} \times \text{area} = 7 \times 5.5179 \approx 38.6253 \, \text{grams}$

Conclusion:

• The area of the machine part is approximately 5.518 square units.
• The mass of the machine part is approximately 38.63 grams.

Would you like more details on the integration steps or visual representation of the curves?

Here are 5 related questions:

1. How does the graph of $\cosh(x)$ differ from $e^x$?
2. How do you calculate the mass of an object with variable density?
3. What are the properties of hyperbolic functions like $\cosh(x)$ and $\sinh(x)$?
4. How do you find the volume of a 3D solid using integration?
5. What is the relationship between area under a curve and definite integrals?

Tip: When integrating, it's helpful to break complex regions into smaller, more manageable integrals.