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Chapter 5: Problem 97

Evaluate. $$ \int_{-1}^{1}\left(1-\sqrt{1-x^{2}}\right) d x $$

Short Answer

Expert verified
2 - \( \frac{\pi}{2} \ \

Step by step solution

01

- Identify the integral

The given integral is \ \ \ \ \ \( \int_{-1}^{1} \left(1-\sqrt{1-x^{2}}\right) dx \ \). This is a definite integral with the limits from \(-1\) to \(1\).
02

- Break the integral into two parts

We can split the integral into two separate integrals: \ \ \ \[ \int_{-1}^{1} 1 \ dx \ - \ \int_{-1}^{1}\sqrt{1-x^{2}} \ dx \ \]
03

- Evaluate the first integral

The first integral is simply: \ \ \ \[ \int_{-1}^{1} 1 \ dx = x \bigg|_{-1}^{1} = 1 - (-1) = 2 \ \]
04

- Evaluate the second integral

The second integral represents the area of a semicircle with radius \(1\). The area of a full circle with radius \(1\) is \( \pi(1)^2 \ = \ \pi \). Therefore, the area of the semicircle is: \ \ \ \[ \int_{-1}^{1} \sqrt{1-x^{2}} \ dx = \frac{\pi}{2} \ \]
05

- Combine the results

Finally, combine the results from the two integrals: \ \ \ \[ \int_{-1}^{1} \left(1-\sqrt{1-x^{2}}\right) \ dx = 2 - \frac{\pi}{2} \ \]

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Key Concepts

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

Integral Evaluation
Evaluating integrals can often seem intimidating, but when broken down step-by-step, it becomes more manageable. A definite integral has both bounds, such as our given integral \( \int_{-1}^{1} \left(1-\sqrt{1-x^{2}}\right) dx \). The goal is to find the area under the curve from \(-1\) to \(1\). It's crucial to understand each step.
Semicircle Area
In calculus, knowing how to interpret geometric shapes helps simplify certain integrals. When evaluating \[ \int_{-1}^{1} \sqrt{1-x^{2}} dx \], we recognize this as the equation of a semicircle. The function \( \sqrt{1-x^{2}} \) represents the upper half of a circle with radius 1 centered at the origin. Hence, finding its integral over \(-1\) to \(1\) gives the area of a semicircle. A full circle with radius 1 has an area \( \pi (1)^2 = \pi \), so half the area would be \(\frac{\pi}{2} \).
Splitting Integrals
To simplify complex integrals, we can split them into smaller, more manageable parts. For example, our integral \[ \int_{-1}^{1} \(1-\sqrt{1-x^{2}}\) dx \], can be divided into two separate integrals: \[ \int_{-1}^{1} 1 \ dx \ - \int_{-1}^{1} \sqrt{1-x^{2}} \ dx \]. Evaluating each part individually: \ \ \ \ \ \( \int_{-1}^{1} 1 \ dx = 2 \), \ \( \int_{-1}^{1} \sqrt{1-x^{2}} \ dx = \frac{\pi}{2} \). Combining these results, we get \[ 2 - \frac{\pi}{2} \]. This method of splitting makes the integration process straightforward.

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