Problem 60 - Applied Calculus

Question

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 2(e^2 - e) \).

Step 1: Understanding the Integral

We need to evaluate the integral \( \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \). This integral can be solved using substitution.

Step 2: Setting up Substitution

Let \( u = \sqrt{x} \) which implies that \( x = u^2 \). Then, the differential \( dx = 2u \, du \). Substituting these into the integral, we change the limits of integration as well. When \( x = 1 \), \( u = 1 \), and when \( x = 4 \), \( u = 2 \).

Step 3: Substitute and Simplify

Substitute the values from Step 2 into the integral:\[ \int_{1}^{2} \frac{e^u}{u} \, (2u \, du) = 2 \int_{1}^{2} e^u \, du \] This simplifies to \( 2 \int_{1}^{2} e^u \, du \).

Step 4: Evaluate the Integral

The integral \( \int_{1}^{2} e^u \, du \) is straightforward:\[ \int_{1}^{2} e^u \, du = \left[ e^u \right]_{1}^{2} = e^2 - e^1 \] Thus, \( 2 \int_{1}^{2} e^u \, du = 2(e^2 - e^1) = 2(e^2 - e) \).

Step 5: Verification with Graphing Calculator

Using a graphing calculator, input \( \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \). Ensure that the calculator settings are correct and ensure the computed area under the curve matches \( 2(e^2 - e) \). If they match, the solution is verified.

Key Concepts

Substitution Method

The substitution method is a powerful tool in solving integrals, especially when the function seems complicated. In this exercise, we solve the integral \( \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \) using substitution, a common integration technique. Here’s a simplified way:

• We look for a substitution that will simplify the integral. Here, we choose \( u = \sqrt{x} \), which transforms our integral into one that's easier to manage.
• Once we set \( u = \sqrt{x} \), \( x \) becomes \( u^2 \), and the differential \( dx \) is replaced with \( 2u \, du \).
• Substitution often requires adjusting the limits of integration. With \( x = 1 \), \( u = 1 \), and when \( x = 4 \), \( u = 2 \).

Replacing these into the integral simplifies our expression considerably, resulting in \( 2 \int_{1}^{2} e^u \, du \). This transformation makes it easier to perform the integration, turning a seemingly complex problem into a more straightforward task.

Integration Techniques

A variety of integration techniques exist to evaluate integrals like the one in this problem. After substitution, we arrive at the integral \( 2 \int_{1}^{2} e^u \, du \). This integral is direct because the antiderivative of \( e^u \) is simply \( e^u \):

• Evaluate \( \int e^u \, du \) to get \( e^u \).
• Apply the fundamental theorem of calculus by substituting the limits of integration back into the equation: \[ e^u \bigg|_{1}^{2} = e^2 - e^1. \]
• Finally, multiply the result by 2 to account for the substitution factor, leading to \( 2(e^2 - e) \).

Integration techniques often involve finding an appropriate method to simplify the integral for easier evaluation. Substitution and understanding basic integrals, such as \( e^u \), are crucial parts of successfully solving these problems.

Graphing Calculator Verification

Verification of manual calculations using technology is a standard practice, especially in calculus. A graphing calculator can be a valuable tool for verifying your solutions in definite integrals. By inputting the original integral \( \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \) into the calculator, students can:

• Ensure they set the calculator with the correct integral expression and limits for accurate results.
• Observe the computed area under the curve, offering a visual understanding of the integral.
• Compare the calculator’s result with their manually derived solution, which in this instance should show \( 2(e^2 - e) \).

By using a graphing calculator, you not only verify the solution but also get visual insights into the behavior of the function over the specified interval. This step is beneficial in understanding the practical application of definite integrals and confirming the precision of your calculations.