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Chapter 5: Problem 56

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{0}^{3} \sqrt{x^{2}+16} x d x $$

Short Answer

Expert verified
The value of the integral is approximately 20.33.

Step by step solution

01

Identify the Integral to Solve

We're given the integral: \( \int_{0}^{3} \sqrt{x^{2}+16} \cdot x \, dx \). Identify the integral as a definite integral with limits 0 and 3.
02

Simplification Using Substitution

Use substitution to simplify the integral. Let \( u = x^2 + 16 \), resulting in \( du = 2x \, dx \), or \( x \, dx = \frac{1}{2} du \). Then, change the limits of integration: when \( x = 0, u = 16 \); when \( x = 3, u = 25 \). The integral becomes \( \frac{1}{2} \int_{16}^{25} \sqrt{u} \, du \).
03

Evaluate the Integral

Evaluate the new integral \( \frac{1}{2} \int_{16}^{25} u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is \( \frac{2}{3}u^{3/2} \). So, the integral becomes \( \frac{1}{2} \left[ \frac{2}{3}u^{3/2} \right]_{16}^{25} = \frac{1}{3} [u^{3/2}]_{16}^{25} \).
04

Solve for Definite Integral Limits

Calculate \( \frac{1}{3} ([25^{3/2}] - [16^{3/2}]) \). \( 25^{3/2} = (25^{1/2})^3 = 5^3 = 125 \) and \( 16^{3/2} = (16^{1/2})^3 = 4^3 = 64 \). Substitute back into the equation \( \frac{1}{3} (125 - 64) = \frac{1}{3} \times 61 = 20.3333 \) (repeating).
05

Verification Using Graphing Calculator

Use a graphing calculator to verify the calculated value of the integral \( \int_{0}^{3} \sqrt{x^{2}+16} \cdot x \, dx \). Graphing calculators provide direct computation for double-checking the result found by hand.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method in Integration
The substitution method is a handy technique used to simplify integrals. It especially comes into play when dealing with integrals that aren't straightforward to solve. The basic idea is to transform the original integral into a simpler one using a substitution.In the given problem, you notice the expression \( \sqrt{x^{2}+16} \cdot x \). Here, choosing a substitution can help in rewriting the function in a more digestible form. Let's dive into how it works:
  • First, identify a part of the integrand that can be set equal to a new variable. In this case, letting \( u = x^2 + 16 \) simplifies the integral because the derivative of \( u \), \( du = 2x \, dx \), matches well with the remaining \( x \, dx \).
  • Solving for \( x \, dx \) gives you \( x \, dx = \frac{1}{2} \, du \), which can substitute back into the integral.
  • The limits of integration must change with respect to the new variable. So when \( x = 0 \), \( u = 16 \) and when \( x = 3 \), \( u = 25 \).
This substitution effectively simplifies the integral to \( \frac{1}{2} \int_{16}^{25} \sqrt{u} \, du \), making it much easier to solve.
Antiderivatives
Antiderivatives play a crucial role in solving integrals. They are essentially the reverse process of differentiation; if you have a function, the antiderivative is a function whose derivative is the original function.For the problem at hand, once you've made the substitution, you are left to solve \( \frac{1}{2} \int_{16}^{25} u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is obtained as follows:
  • The general rule for finding the antiderivative \( \int u^n \, du \) is \( \frac{u^{n+1}}{n+1} + C \) where \( C \) is a constant."
  • Applying this rule, you get \( \frac{2}{3} u^{3/2} \) as the antiderivative of \( u^{1/2} \).
Using the definite integral limits, calculate \( \frac{1}{3} \left[ u^{3/2} \right]_{16}^{25} \). Evaluating it, you substitute 25 and 16 into the antiderivative:
  • \( 25^{3/2} = 125 \) and \( 16^{3/2} = 64 \).
  • Finally, compute \( \frac{1}{3} (125 - 64) = 20.3333 \).
By following these steps, you can consistently calculate definite integrals using antiderivatives.
Graphical Verification of Integrals
Verifying the results of integration is vital to ensure accuracy. One of the most intuitive techniques is checking the solution using a graphing calculator. Graphical verification helps in comparing the analytical results you calculated by hand, with a numerically computed value.Here鈥檚 how this can be tackled in the context of our problem:
  • After calculating the solution manually, input the integral into a graphing calculator: \( \int_{0}^{3} \sqrt{x^{2}+16} x \, dx \).
  • The calculator will compute the definite integral and provide a result.
  • Compare this graphed value with the manual answer, which was approximately 20.3333. A match confirms your solution鈥檚 accuracy.
Graphing calculators not only offer a numerical check but also visually display the area under the curve, which represents the definite integral's value. Seeing this aids in understanding how integral values reflect areas in geometry.

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