Question

Given the integral \(\int_{0}^{1} \sqrt{1-x^{2}} d x\). Make the substitution \(x=\sin t .\) Is it possible to take the numbers \(\pi\) and \(\frac{\pi}{2}\) as the limits for \(t ?\)

Short answer

In conclusion, when using the substitution \(x = \sin t\) to evaluate the integral \(\int_{0}^{1} \sqrt{1-x^{2}} d x\), we can indeed take the numbers \(0\) and \(\frac{\pi}{2}\) as the limits for \(t\). The transformed integral becomes \(\int_{0}^{\frac{\pi}{2}} \cos^3 t d t\).

Step-by-step solution

Make the substitution

Given \(x = \sin t\), we will first find \(\frac{dx}{dt}\): \(\frac{d x}{d t}=\frac{d(\sin t)}{d t}=\cos t\) From the given integral \(\int_{0}^{1} \sqrt{1-x^{2}} d x\) substitute \(x=\sin t\) and \(dx = \cos t dt\): \(\int \sqrt{1-\sin^2 t} \cos t d t\) Using the identity \(\sin^2 t + \cos^2 t = 1\), we get: \(\int \cos^2 t \cdot \cos t d t\) \(=\int \cos^3 t d t\)

Find the new limits for \(t\)

We're given the limits of \(x\) as \(0\) and \(1\). Now we need to find the corresponding limits for \(t\): For the lower limit \(x=0\), we have: \(0 = \sin t\) The value of \(t = 0\) satisfies this equation. For the upper limit \(x=1\), we have: \(1 = \sin t\) The value of \(t = \frac{\pi}{2}\) satisfies this equation. So, the new limits for the integral are \(0\) and \(\frac{\pi}{2}\).

Check if it's possible to take the given limits for \(t\)

We have found the new limits for \(t\) to be \(0\) and \(\frac{\pi}{2}\). These limits are in fact exactly the same as the ones provided in the question. Therefore, it is possible to take the numbers \(\pi\) and \(\frac{\pi}{2}\) as the limits for \(t.\)

Key concepts

Trigonometric Substitution

Trigonometric substitution is an integration technique that is especially useful when dealing with square roots of expressions like 1 minus the square of a variable, expressions that remind us of the Pythagorean identity. In the given exercise, the substitution involves changing the variable/expression, such as changing the variable from 'x' to 't' using a trigonometric function, in this case, a sine function.

By making the substitution of x with sin(t), the algebraic expression inside the root becomes a trigonometric identity. This simplifies the integration process by utilizing known integrals of trigonometric functions. The technique also requires recalculating the integration limits based on the substitution, ensuring we're evaluating the integral over the correct interval.

Definite Integral

A definite integral has both a start and an end point, which are referred to as integration limits. These limits specify the region over which the function is integrated. In particular, for the integral \( \int_{a}^{b} f(x) dx \), 'a' and 'b' represent the lower and upper limits, respectively.

When you calculate a definite integral, you are essentially finding the net area under the curve of the function from 'a' to 'b'. The definite integral provides a numerical value, unlike the indefinite integral which represents a family of functions or antiderivatives with the addition of the constant of integration.

Integration Limits

When applying a substitution in integration, the original integration limits need to be changed to match the new variable. This ensures that the area calculated under the curve of the new function corresponds exactly to the area under the original curve between the original limits.

As shown in the solution, when substituting x with sin(t), the limits for 'x' being 0 and 1 are translated into limits for 't' using the inverse of the substitution function. It's crucial to make this conversion to maintain the integrity of the definite integral. Without this step, the result of the integral would be incorrect.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are useful tools in simplifying expressions and solving equations that involve trigonometric functions.

One of the most basic and frequently used identities is \( \sin^{2}t + \cos^{2}t = 1 \), which was applied in our problem. By applying this identity, the integral which originally had a square root of \(1-x^2\) becomes an integral with \(\cos^3 t\), a simpler expression that is easier to integrate. Trigonometric identities thus enable the transformation of the given problem into a form that is readily solvable with standard integration techniques.