Evaluate The Definite Integral. 1 3 1 7x Dx 0

Evaluating definite integrals might seem daunting at first, but it's a fundamental concept in calculus with applications ranging from physics to economics. The definite integral represents the signed area under a curve between two specified limits. In this case, we'll focus on evaluating the definite integral of the function f(x) = 7x from the lower limit of 0 to the upper limit of 3. This process involves understanding the concept of antiderivatives, applying the Fundamental Theorem of Calculus, and performing basic arithmetic.

Understanding Definite Integrals

Before diving into the specific evaluation, it's crucial to grasp the core ideas behind definite integrals. Integration, at its heart, is the reverse process of differentiation. While differentiation finds the rate of change of a function, integration finds the accumulation of a function over an interval.

• Antiderivative: The antiderivative of a function f(x) is a function F(x) whose derivative is f(x). In other words, F'(x) = f(x). For instance, the antiderivative of x is (1/2)*x^2 + C, where C is the constant of integration.
• Definite Integral: A definite integral has upper and lower limits of integration. It represents the net signed area between the function's curve and the x-axis within those limits.
• Fundamental Theorem of Calculus: This theorem connects differentiation and integration. It states that if F(x) is the antiderivative of f(x), then the definite integral of f(x) from a to b is F(b) - F(a).

Evaluating the Definite Integral of ∫₀³ 7x dx

Now, let's apply these concepts to evaluate the given definite integral: ∫₀³ 7x dx. This notation means we want to find the definite integral of the function 7x with respect to x, from the lower limit of integration 0 to the upper limit of integration 3.

Step 1: Find the Antiderivative

The first step is to find the antiderivative of the function f(x) = 7x. To do this, we can use the power rule for integration, which states that the integral of xⁿ is (x^(n+1))/(n+1) + C, where n ≠ -1 and C is the constant of integration.

Applying this rule to 7x:

• Rewrite 7x as 7x¹.
• Increase the exponent by 1: 1 + 1 = 2.
• Divide by the new exponent: 7x² / 2.
• Add the constant of integration, C: (7/2)x² + C.

Therefore, the antiderivative of 7x is (7/2)x² + C. We can denote this as F(x) = (7/2)x² + C.

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that to evaluate the definite integral, we need to find the difference between the antiderivative evaluated at the upper limit and the antiderivative evaluated at the lower limit. In this case, the upper limit is 3 and the lower limit is 0.

So, we need to calculate F(3) - F(0).

• F(3): Substitute x = 3 into the antiderivative:
  • F(3) = (7/2)(3)² + C = (7/2)(9) + C = 63/2 + C
• F(0): Substitute x = 0 into the antiderivative:
  • F(0) = (7/2)(0)² + C = (7/2)(0) + C = 0 + C = C

Now, subtract F(0) from F(3):

• F(3) - F(0) = (63/2 + C) - (C) = 63/2

Notice that the constant of integration, C, cancels out. This is always the case when evaluating definite integrals, so we often omit writing C when finding the antiderivative for definite integral calculations.

Step 3: Simplify the Result

The result, 63/2, is a fraction. We can express it as a mixed number or a decimal:

• Mixed Number: 63/2 = 31 1/2
• Decimal: 63/2 = 31.5

Therefore, the value of the definite integral ∫₀³ 7x dx is 63/2, or 31.5. This means the net signed area under the curve y = 7x from x = 0 to x = 3 is 31.5 square units. Since the function 7x is always positive in the interval [0, 3], the area is entirely above the x-axis.

A More Concise Calculation

We can perform the calculation more concisely after finding the antiderivative:

1. Find the antiderivative: F(x) = (7/2)x² (We can omit the constant of integration, C).
2. Evaluate at the upper and lower limits:
   • F(3) = (7/2)(3)² = (7/2)(9) = 63/2
   • F(0) = (7/2)(0)² = 0
3. Subtract: F(3) - F(0) = 63/2 - 0 = 63/2 = 31.5

Visualizing the Integral

The function f(x) = 7x represents a straight line passing through the origin with a slope of 7. The definite integral ∫₀³ 7x dx calculates the area of the triangle formed by this line, the x-axis, and the vertical line x = 3.

• Base of the triangle: The base is the interval along the x-axis from 0 to 3, so the base length is 3.
• Height of the triangle: The height is the value of the function at x = 3, which is f(3) = 7(3) = 21.
• Area of the triangle: Area = (1/2) * base * height = (1/2) * 3 * 21 = 63/2 = 31.5

This geometric interpretation confirms our calculated value of the definite integral. It highlights that the definite integral, in this case, corresponds directly to the area of a simple geometric shape.

Properties of Definite Integrals

Understanding some key properties of definite integrals can simplify calculations and provide insights:

• Linearity: ∫ₐᵇ [cf(x) + dg(x)] dx = c∫ₐᵇ f(x) dx + d∫ₐᵇ g(x) dx, where c and d are constants. We used this implicitly by factoring out the 7 when finding the antiderivative of 7x.
• Reversing Limits: ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx. Changing the order of the limits of integration changes the sign of the integral.
• Additivity: ∫ₐᶜ f(x) dx = ∫ₐᵇ f(x) dx + ∫ᵇᶜ f(x) dx, where a < b < c. We can break up the interval of integration.
• Integral of Zero: ∫ₐᵃ f(x) dx = 0. The integral from a point to itself is always zero.
• Constant Multiple: ∫ₐᵇ cf(x) dx = c∫ₐᵇ f(x) dx, where c is a constant. This is useful for simplifying integrals with constant coefficients.

Applications of Definite Integrals

Definite integrals are powerful tools with numerous applications in various fields:

• Physics: Calculating displacement, work done by a force, and the center of mass of an object. For example, if v(t) represents the velocity of an object at time t, then ∫ₐᵇ v(t) dt represents the displacement of the object between times a and b.
• Engineering: Determining areas, volumes, and moments of inertia, which are crucial for structural design and analysis.
• Economics: Calculating consumer surplus, producer surplus, and present value of future income streams.
• Probability and Statistics: Finding probabilities associated with continuous random variables and calculating expected values. The area under a probability density function (PDF) between two points represents the probability that the random variable falls within that range.
• Computer Graphics: Calculating areas of complex shapes and performing image processing operations.

Common Mistakes to Avoid

When evaluating definite integrals, be mindful of these common mistakes:

• Forgetting the Constant of Integration (for Indefinite Integrals): While the constant of integration cancels out in definite integrals, it's essential to include it when finding the indefinite integral (the general antiderivative).
• Incorrectly Applying the Power Rule: Ensure you correctly increase the exponent and divide by the new exponent. Remember the power rule doesn't apply when n = -1.
• Reversing the Order of Subtraction: Always subtract the value of the antiderivative at the lower limit from the value at the upper limit: F(b) - F(a). Reversing this order will result in the negative of the correct answer.
• Ignoring the Sign of the Function: Definite integrals represent the net signed area. If the function is negative over part of the interval, the integral will account for that negative area.
• Not Simplifying: Always simplify the result of the integration to obtain the final answer in its simplest form.

Examples of Evaluating Other Definite Integrals

Let's look at a few more examples to solidify our understanding:

Example 1: ∫₁² x² dx

1. Antiderivative: The antiderivative of x² is (1/3)x³.
2. Evaluate at limits:
   • (1/3)(2)³ = 8/3
   • (1/3)(1)³ = 1/3
3. Subtract: (8/3) - (1/3) = 7/3

Therefore, ∫₁² x² dx = 7/3.

Example 2: ∫₀^(π/2) cos(x) dx

1. Antiderivative: The antiderivative of cos(x) is sin(x).
2. Evaluate at limits:
   • sin(π/2) = 1
   • sin(0) = 0
3. Subtract: 1 - 0 = 1

Therefore, ∫₀^(π/2) cos(x) dx = 1.

Example 3: ∫₋₁¹ e^x dx

1. Antiderivative: The antiderivative of e^x is e^x.
2. Evaluate at limits:
   • e¹ = e
   • e⁻¹ = 1/e
3. Subtract: e - (1/e)

Therefore, ∫₋₁¹ e^x dx = e - (1/e).

Advanced Integration Techniques

While we've covered basic definite integrals, many functions require more advanced techniques to integrate. Some common techniques include:

• U-Substitution: Used when the integral involves a composite function and its derivative. This technique simplifies the integral by substituting a part of the function with a new variable, u.
• Integration by Parts: Used to integrate the product of two functions. This technique is based on the product rule of differentiation and involves choosing appropriate "u" and "dv" parts of the integrand.
• Trigonometric Substitution: Used for integrals involving square roots of quadratic expressions. This technique involves substituting trigonometric functions for the variable of integration.
• Partial Fraction Decomposition: Used to integrate rational functions (ratios of polynomials). This technique involves breaking down the rational function into simpler fractions that are easier to integrate.
• Improper Integrals: Integrals where one or both limits of integration are infinite, or the function has a discontinuity within the interval of integration. These integrals require special techniques to evaluate their convergence.

Conclusion

Evaluating definite integrals is a core skill in calculus with wide-ranging applications. By understanding the fundamental concepts of antiderivatives, the Fundamental Theorem of Calculus, and mastering basic integration techniques, you can confidently solve a variety of integration problems. Remember to pay attention to detail, avoid common mistakes, and practice regularly to build your proficiency. The ability to evaluate definite integrals empowers you to analyze and solve problems in diverse fields, from physics and engineering to economics and statistics. And, as we saw, the simple definite integral ∫₀³ 7x dx neatly calculates the area under a line, highlighting the power and elegance of calculus.