Problem 316
Question
The area of a semicircle of radius 1 can be expressed as \(\int_{-1}^{1} \sqrt{1-x^{2}} d x .\) Use the substitution \(x=\cos t\) to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral.
Step-by-Step Solution
Verified Answer
The modified integral is \( \int_{0}^{\pi} \sin^2 t \, dt \).
1Step 1: Identify the substitution
We are given the substitution \( x = \cos t \). From this we have \( dx = -\sin t \, dt \). This change of variables will allow us to rewrite the original integral in terms of \( t \).
2Step 2: Change the limits of integration
Since \( x = -1 \) and \( x = 1 \) correspond to the endpoints of the semicircle, translate these into \( t \) values. We find that when \( x = -1 \), \( \cos t = -1 \), so \( t = \pi \). Similarly, for \( x = 1 \), \( \cos t = 1 \), so \( t = 0 \).
3Step 3: Substitute and rewrite the integral
Substitute \( x = \cos t \) and \( dx = -\sin t \, dt \) into the integral. This gives us:\[\int_{-1}^{1} \sqrt{1-x^2} \, dx = \int_{\pi}^{0} \sqrt{1-\cos^2 t} (-\sin t) \, dt.\]
4Step 4: Simplify the integral expression
Realize that \( \sqrt{1-\cos^2 t} = \sin t \), so the integral becomes:\[ \int_{\pi}^{0} \sin t (-\sin t) \, dt = \int_{\pi}^{0} -\sin^2 t \, dt. \]
5Step 5: Adjust integral limits and sign
Change the limits and the sign of the integral to maintain the same evaluation direction. This gives \[ \int_{0}^{\pi} \sin^2 t \, dt. \]
Key Concepts
Trigonometric SubstitutionSemicircle Area Using IntegralsIntegral Transformation
Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals, especially when the expression involves a square root of a quadratic expression. The core idea is to replace a variable with a trigonometric function to take advantage of trigonometric identities. This can transform a complex expression into a simpler one, facilitating integration.
In the exercise, we applied the substitution technique by setting \( x = \cos t \). This step involves replacing the variable \( x \) with \( \cos t \) and finding the differential \( dx = -\sin t \, dt \). This substitution is suitable because it makes use of the Pythagorean identity \( 1 - \cos^2 t = \sin^2 t \) to simplify the radical expression \( \sqrt{1-x^2} \).
There are common trigonometric substitutions, such as:
In the exercise, we applied the substitution technique by setting \( x = \cos t \). This step involves replacing the variable \( x \) with \( \cos t \) and finding the differential \( dx = -\sin t \, dt \). This substitution is suitable because it makes use of the Pythagorean identity \( 1 - \cos^2 t = \sin^2 t \) to simplify the radical expression \( \sqrt{1-x^2} \).
There are common trigonometric substitutions, such as:
- \( x = a \sin \theta \) for \( \sqrt{a^2 - x^2} \)
- \( x = a \tan \theta \) for \( \sqrt{a^2 + x^2} \)
- \( x = a \sec \theta \) for \( \sqrt{x^2 - a^2} \)
Semicircle Area Using Integrals
The semicircle area can be calculated using integrals by considering its geometric properties. The formula \( \int_{-1}^{1} \sqrt{1-x^{2}} \, dx \) is derived from the equation of a circle, \( x^2 + y^2 = 1 \), where \( y = \sqrt{1-x^2} \) represents the top half, or positive square root of the circle.
This integral computes the area of a semicircle from \( x = -1 \) to \( x = 1 \), essentially covering the top half of the unit circle. The result represents half of the total area of a circle with a given radius. To solve this with integration, we transform the integral to a more manageable form, typically using substitutions like the given \( x = \cos t \).
This transformation is particularly effective as it aligns the limits of integration with familiar trigonometric values, i.e., \( t = \pi \) and \( t = 0 \) correspond to \( x = -1 \) and \( x = 1 \) respectively. This allows us to compute complex geometric areas thoroughly through calculus concepts.
This integral computes the area of a semicircle from \( x = -1 \) to \( x = 1 \), essentially covering the top half of the unit circle. The result represents half of the total area of a circle with a given radius. To solve this with integration, we transform the integral to a more manageable form, typically using substitutions like the given \( x = \cos t \).
This transformation is particularly effective as it aligns the limits of integration with familiar trigonometric values, i.e., \( t = \pi \) and \( t = 0 \) correspond to \( x = -1 \) and \( x = 1 \) respectively. This allows us to compute complex geometric areas thoroughly through calculus concepts.
Integral Transformation
Integral transformation is a powerful method in calculus for converting integrals into an equivalent form that is easier to solve or interpret. Through substitution, the original integral bounds, and expression, are transformed into a more simple form.
In our exercise, the substitution \( x = \cos t \) transforms the complicated square root integral into a manageable trigonometric one. The original integral \( \int_{-1}^{1} \sqrt{1-x^2} \, dx \) is transformed into \( \int_{0}^{\pi} \sin^2 t \, dt \).
Why do we transform? Because solving the transformed integral is often easier. Here, \( \sin^2 t \) is easier to integrate than \( \sqrt{1-x^2} \). Plus, trigonometric integrals allow replacement using identities and known antiderivatives. Transformations help maintain the problem's inherent geometric meaning while simplifying mathematical complexity.
The key steps involved in integral transformation include:
In our exercise, the substitution \( x = \cos t \) transforms the complicated square root integral into a manageable trigonometric one. The original integral \( \int_{-1}^{1} \sqrt{1-x^2} \, dx \) is transformed into \( \int_{0}^{\pi} \sin^2 t \, dt \).
Why do we transform? Because solving the transformed integral is often easier. Here, \( \sin^2 t \) is easier to integrate than \( \sqrt{1-x^2} \). Plus, trigonometric integrals allow replacement using identities and known antiderivatives. Transformations help maintain the problem's inherent geometric meaning while simplifying mathematical complexity.
The key steps involved in integral transformation include:
- Choosing an appropriate substitution to simplify the function
- Changing the limits of integration accordingly
- Using identities to simplify the new integral expression
Other exercises in this chapter
Problem 314
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View solution Problem 320
In the following exercises, compute each indefinite integral. $$\int e^{2 x} d x$$
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