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Magazine Chapter 28: Problem 39
Solve the given problems by integration. Find the area bounded by \(y=\sin ^{4} x, y=\cos ^{4} x,\) and \(x=0\) in the first quadrant.
Short Answer
Expert verified
The area is \(\frac{1}{2}\).
Step by step solution
01
Understand the Problem
We need to find the area in the first quadrant that is bounded by the curves \(y = \sin^4 x\) and \(y = \cos^4 x\) from \(x = 0\) to their intersection point.
02
Find the Intersection Points
Since the area is bound by the points where \(\sin^4 x = \cos^4 x\), solve the equation: \(\sin^4 x = \cos^4 x\). This simplifies to \(\sin^2 x = \cos^2 x\) by taking the square root of both sides twice. Finally, \(\tan^2 x = 1\) suggests \(x = \frac{\pi}{4}\). Thus, integration bounds are from \(x=0\) to \(x = \frac{\pi}{4}\).
03
Set Up the Integral for Area Calculation
The area between two curves from \(x = 0\) to \(x = \frac{\pi}{4}\) is given by the integral \(\int_0^{\pi/4} (\cos^4 x - \sin^4 x)\, dx\).
04
Simplify the Integral Expression
Use the identity \(\cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\). Since \(\cos^2 x + \sin^2 x = 1\), it simplifies to \(\cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x = \cos 2x\).
05
Integrate the Function
Now, compute the integral \(\int_0^{\pi/4} \cos 2x \, dx\). The antiderivative of \(\cos 2x\) is \(\frac{1}{2} \sin 2x\). Evaluate it from \(0\) to \(\frac{\pi}{4}\):\[ \frac{1}{2} [\sin(2 \times \frac{\pi}{4}) - \sin(0)] = \frac{1}{2} [\sin(\frac{\pi}{2}) - 0] = \frac{1}{2} [1] = \frac{1}{2}. \]
06
Conclude the Solution
The area bounded by the given curves \(y = \sin^4 x\) and \(y = \cos^4 x\) in the first quadrant from \(x=0\) to \(x=\frac{\pi}{4}\) is \(\frac{1}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Integration
Integration is a fundamental concept in calculus used to find areas under curves. In this exercise, we're interested in the area between two specific curves:
- \(y = \sin^4 x\)
- \(y = \cos^4 x\)
The Role of Trigonometric Identities
Trigonometric identities are relationships between trigonometric functions that simplify expressions. For example, when working with \(\cos^4 x - \sin^4 x\), we rely on the identity \(\cos^2 x - \sin^2 x = \cos 2x\), along with \(\cos^2 x + \sin^2 x = 1\) to simplify. These identities help transform complicated integrals into ones that are easier to evaluate. By converting the difference of squares \(\cos^4 x - \sin^4 x\) using these identities, we make integration more straightforward. This simplification is essential, especially when calculating areas between curves.
Evaluating a Definite Integral
The definite integral helps find the exact area between two points. For our problem, it means calculating \(\int_0^{\pi/4} \cos 2x \, dx\). This involves finding the antiderivative, which must be evaluated at the upper and lower bounds.
The antiderivative of \(\cos 2x\) is \(\frac{1}{2} \sin 2x\). We then evaluate this from 0 to \(\frac{\pi}{4}\):
The antiderivative of \(\cos 2x\) is \(\frac{1}{2} \sin 2x\). We then evaluate this from 0 to \(\frac{\pi}{4}\):
- Calculate \(\frac{1}{2} [\sin(2 \times \frac{\pi}{4}) - \sin(0)]\)
- Complete the subtraction to find the difference between the sine values at these endpoints.
Insights About the First Quadrant
The first quadrant is the top-right section of the coordinate plane where both x and y values are positive. In trigonometric terms within the first quadrant:
- Functions like sine and cosine yield positive results due to the positive nature of angles located here.
- This makes the first quadrant a common area of analysis where trigonometric values are straightforward and well understood.
