Problem 72 - Thomas Calculus

Question

Find the area between the curve \(y=\tan x\) and the \(x\) -axis from \(x=-\pi / 4\) to \(x=\pi / 3 .\)

Step-by-Step Solution

Answer

The area is \( \frac{1}{2} \ln 2 \).

Step 1: Understanding the Problem

We need to find the area between the curve \( y = \tan x \) and the \( x \)-axis from \( x = -\pi/4 \) to \( x = \pi/3 \). This requires calculating a definite integral of \( \tan x \).

Step 2: Integrate the Function

The integral we need to evaluate is \( \int \tan x \, dx \). By using the identity \( \tan x = \frac{\sin x}{\cos x} \) and knowing its integral formula, we have:\[ \int \tan x \, dx = -\ln |\cos x| + C \] where \( C \) is the constant of integration.

Step 3: Apply Limits to Definite Integral

Now, we apply the definite integral limits from \( x = -\pi/4 \) to \( x = \pi/3 \):\[ \int_{-\pi/4}^{\pi/3} \tan x \, dx = \left. -\ln |\cos x| \right|_{-\pi/4}^{\pi/3}. \]

Step 4: Evaluate at Upper Limit \( x = \pi/3 \)

Calculate \( -\ln |\cos(\pi/3)| \):\[ \cos(\pi/3) = \frac{1}{2} \]Thus, the evaluation at the upper limit is:\[ -\ln \left(\frac{1}{2} \right) = \ln 2 \].

Step 5: Evaluate at Lower Limit \( x = -\pi/4 \)

Calculate \( -\ln |\cos(-\pi/4)| \):\[ \cos(-\pi/4) = \frac{\sqrt{2}}{2} \]Thus, the evaluation at the lower limit is:\[ -\ln \left( \frac{\sqrt{2}}{2} \right) = \frac{1}{2} \ln 2 \].

Step 6: Compute the Area

Subtract the evaluated value at the lower limit from the evaluated value at the upper limit:\[ \ln 2 - \frac{1}{2} \ln 2 = \frac{1}{2} \ln 2 \].

Step 7: Final Answer

The area between the curve \( y = \tan x \) and the \( x \)-axis from \( x = -\pi/4 \) to \( x = \pi/3 \) is given by \( \frac{1}{2} \ln 2 \).

Key Concepts

Definite Integral

A definite integral is a key concept in calculus used to find the accumulated quantity, like area under a curve, from one point to another on the x-axis. It is expressed as \[ \int_{a}^{b} f(x) \, dx \], where \( a \) and \( b \) are the lower and upper limits of integration, respectively. Unlike indefinite integrals, definite integrals result in a numerical value that represents this total accumulation.

To compute a definite integral, follow these steps:

• Find an antiderivative (indefinite integral) of the function \( f(x) \).
• Evaluate this antiderivative at the upper limit \( b \) and lower limit \( a \).
• Subtract the lower limit evaluation from the upper limit evaluation: \[ F(b) - F(a) \].

In the exercise, we used the definite integral of \( \tan x \) over the limits \( x = -\pi/4 \) to \( x = \pi/3 \) to find the area. The definite integral calculated gives us a specific value, reflecting the area between \( y = \tan x \) and the x-axis within those bounds.

Trigonometric Integration

Trigonometric integration involves integrating functions that include trigonometric expressions. Such functions often appear in many calculus problems, requiring us to utilize identities to simplify and solve the integrals. For example, integrating \( \tan x \) directly involves using the identity \( \tan x = \frac{\sin x}{\cos x} \), which allows us to rewrite the integral in a more manageable form.

The integral of \( \tan x \) is \(-\ln |\cos x| + C \), derived from the identity \( \tan x = \frac{\sin x}{\cos x} \) and recognizing the derivative of \( \ln |\cos x| \) involves \( \sec^2 x \), which is related to \( \tan x \) through trigonometric derivatives.

Understanding these identities and their derivatives is vital:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( \frac{d}{dx}[\sin x] = \cos x \)
• \( \frac{d}{dx}[\cos x] = -\sin x \)
• \( \frac{d}{dx}[\tan x] = \sec^2 x \)

These principles facilitate solving integrals involving trigonometric functions, as showcased in the exercise.

Limits of Integration

Limits of integration are crucial in determining the range for accumulation or analysis when calculating a definite integral. They specify the interval over which we calculate the area under a curve.

• The lower limit (\( a \)) indicates where the calculation starts on the x-axis.
• The upper limit (\( b \)) indicates where the calculation ends on the x-axis.

The limits are denoted in the notation \( \int_{a}^{b} \), where \( b \) is greater than \( a \). In our specific exercise, the area between \( y = \tan x \) and the x-axis was calculated between the limits \( x = -\pi/4 \) and \( x = \pi/3 \).

Applying these limits accurately involves:

• Evaluating the antiderivative at the upper limit.
• Evaluating the antiderivative at the lower limit.
• Subtracting the result from the lower limit from the result at the upper limit: \( F(b) - F(a) \).

Every selected range provides a different accumulated quantity, emphasizing the importance of correctly determining and applying limits in definite integrals.