Problem 41 - Calculus: Early Transcendentals

Question

Evaluate \( \displaystyle \int^1_1 \sqrt{1 + x^4}\, dx \).

Step-by-Step Solution

Answer

The integral evaluates to 0 because the limits of integration are identical.

Step 1: Understanding the Exercise

The exercise requires evaluating the definite integral \( \int^1_1 \sqrt{1 + x^4} \, dx \). A definite integral calculates the area under a curve from one point to another. Here, the bounds are from 1 to 1.

Step 2: Realize the Properties of Definite Integrals

One important property of definite integrals is that if the lower limit of integration is equal to the upper limit, then the integral is zero, regardless of the function. This is because you are essentially measuring the area over an interval of zero width.

Step 3: Applying the Property

Apply the property to our given integral: \( \int^1_1 \sqrt{1 + x^4} \, dx = 0 \). Since the limits of integration are the same, the integral evaluates to zero without needing to perform any further calculations.

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus. They help us determine the accumulation of quantities, such as areas under a curve or total displacement.
A definite integral is written as \( \int_a^b f(x) \, dx \), where \(a\) and \(b\) are the lower and upper bounds, respectively.
It gives the net area between the curve \(f(x)\) and the x-axis from the point \(x=a\) to \(x=b\).

• When \(f(x)\) is above the x-axis, the area is considered positive.
• If \(f(x)\) falls below the x-axis, the area is negative.

The definitive part means the result is a specific number rather than a function as in indefinite integrals.
Definite integrals serve many purposes in both pure and applied mathematics, from calculating area to solving real-world problems.

Properties of Integration

The properties of integrals allow for more efficient and powerful solving techniques.
One key property is the effect of equal bounds: if the limits of integration are the same, the integral's value is zero. This is because no interval exists between the boundaries.
For the integral \( \int_a^a f(x) \, dx \), whether the function is complex or simple, it evaluates to zero.
Other important properties of integration include:

• Linearity: The integral of a sum is the sum of the integrals.
• Additivity: Integrals can be added over adjacent intervals.
• Reversal of Limits: Flipping limits changes the sign of the integral \( \int_b^a f(x) \, dx = -\int_a^b f(x) \, dx \).

Knowing these properties simplifies calculations and helps assess whether you're progressing correctly.

Area Under a Curve

Calculating the area under a curve is a central application of definite integrals. This area can represent a multitude of physical quantities:

• Distance or displacement in physics.
• Total growth or change in economics or biology.
• Accumulated value or cost over time in finance.

Using a definite integral \( \int_a^b f(x) \, dx \), you can find the total area between the curve \(f(x)\) and the x-axis between the specified bounds.

For the interval \([a,b]\), the concept assumes that the curve is well-behaved—meaning it has no breaks or sharp corners.
If \(f(x)\) changes sign, consider splitting the integral at these points to accurately account for the net area.
Understanding this concept provides insight into both practical applications and theoretical explorations in calculus.