Problem 8 - Thomas Calculus

Question

In Exercises \(1-8 :\) a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface, too. c. Use your grapher's or computer's integral evaluator to find the surface's area numerically. \(y=\int_{1}^{x} \sqrt{t^{2}-1} d t, \quad 1 \leq x \leq \sqrt{5} ; \quad x\) -axis

Step-by-Step Solution

Answer

The surface area can be found by evaluating the integral \( S = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^2 - 1} \, dt \right) x \, dx \) numerically.

Step 1: Understand the Problem

We want to find the surface area generated by revolving the curve defined by the integral \( y = \int_{1}^{x} \sqrt{t^2 - 1} \, dt \) about the \( x \)-axis over the interval \( 1 \leq x \leq \sqrt{5} \).

Step 2: Identify the Surface Area Formula

The formula for the surface area \( S \) of a curve \( y = f(x) \) revolving around the \( x \)-axis from \( x = a \) to \( x = b \) is \( S = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \).

Step 3: Differentiate the Curve

The curve given is \( y = \int_{1}^{x} \sqrt{t^2 - 1} \, dt \). Differentiating it with respect to \( x \), we get \( \frac{dy}{dx} = \sqrt{x^2 - 1} \) using the Fundamental Theorem of Calculus.

Step 4: Substitute into Surface Area Formula

Substitute \( y = \int_{1}^{x} \sqrt{t^2 - 1} \, dt \) and \( \frac{dy}{dx} = \sqrt{x^2 - 1} \) into the surface area formula: \[ S = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^2 - 1} \, dt \right) \sqrt{1 + (x^2 - 1)} \, dx \] Simplify the term under the square root: \( \sqrt{1 + (x^2 - 1)} = x \).

Step 5: Simplify the Integral

Plug the simplification back into the integral: \[ S = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^2 - 1} \, dt \right) x \, dx \] This is the integral that needs to be computed to find the surface area numerically.

Step 6: Graph the Curve and Surface

Graph the curve \( y = \int_{1}^{x} \sqrt{t^2 - 1} \, dt \) to visualize it over \( 1 \leq x \leq \sqrt{5} \). Graphing software can also generate the shape of the surface formed by revolving this curve around the \( x \)-axis.

Step 7: Evaluate the Integral Numerically

Use a graphing calculator or computer software to evaluate the integral numerically. This will give you the area of the surface created when the curve is revolved around the \( x \)-axis over the given interval.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a crucial concept that links the process of differentiation with integration. It provides a method to evaluate definite integrals by recognizing the antiderivative. Here's how it applies to the exercise:

• Consider the curve given by the definite integral \( y = \int_{1}^{x} \sqrt{t^2 - 1} \, dt \). This defines a new function \( y \) which is an antiderivative of \( \sqrt{x^2 - 1} \).
• By the Fundamental Theorem of Calculus, we can differentiate the function \( y \) with respect to \( x \) to obtain \( \frac{dy}{dx} = \sqrt{x^2 - 1} \).

In simple terms, this theorem allows us to transition from the accumulated area under a curve to the rate of change of the curve itself. In our exercise, using the theorem simplifies the steps to find the differential needed for the surface area formula.

Numerical Integration

Numerical Integration is an approach used when finding an exact solution of an integral is complex or impossible. The exercise involves evaluating a challenging integral to calculate the surface area, which often benefits from numerical methods.

• The integral to compute is \( S = 2\pi \int_{1}^{\sqrt{5}} \left( \int_{1}^{x} \sqrt{t^2 - 1} \, dt \right) x \, dx \). Given its complexity, we rely on numerical tools.
• Graphing calculators or computational software use methods like Simpson's rule, Trapezoidal rule, or other algorithms to approximate the result effectively.

These techniques find approximate solutions by dividing the area into simpler sub-areas (like trapezoids or rectangles) and summing them up. It's handy because it makes dealing with complicated integrals more approachable, allowing us to compute the surface area without an explicit antiderivative.

Graphing Techniques

Visualizing mathematical concepts makes comprehension simpler and aids in error-checking. Graphing the curve and the surface it forms upon revolution offers a geometric perspective.

• For the curve \( y = \int_{1}^{x} \sqrt{t^2 - 1} \, dt \), graphing software can plot this function across the interval \( 1 \leq x \leq \sqrt{5} \).
• The revolving of this curve around the \( x \)-axis forms a surface, which can also be graphically represented with specialized tools, showcasing the three-dimensional aspect.

These graphing techniques help visualize the geometry involved, providing insights into the behavior of the function and ensuring that calculations are more intuitive and less abstract. It's like seeing the problem unfold in real-life dimensions, making it more interactive and less prone to errors.