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Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The image depicts a definite integral, which is an important concept in calculus. The integral is expressed as follows:

\[ \int_{0}^{\frac{\pi}{2}} \sin^7 \theta \cos^5 \theta \, d\theta \]

This notation represents the integral of the function \(\sin^7 \theta \cos^5 \theta\) with respect to \(\theta\), evaluated from \(\theta = 0\) to \(\theta = \frac{\pi}{2}\). Integrals like this are often used to calculate areas under curves, among other applications in mathematics and engineering. 

The integrand, \(\sin^7 \theta \cos^5 \theta\), indicates that both sine and cosine functions are raised to powers and multiplied together within the integral. Solving this type of integral often involves using trigonometric identities or substitution methods to simplify the expression.
Transcribed Image Text:The image depicts a definite integral, which is an important concept in calculus. The integral is expressed as follows: \[ \int_{0}^{\frac{\pi}{2}} \sin^7 \theta \cos^5 \theta \, d\theta \] This notation represents the integral of the function \(\sin^7 \theta \cos^5 \theta\) with respect to \(\theta\), evaluated from \(\theta = 0\) to \(\theta = \frac{\pi}{2}\). Integrals like this are often used to calculate areas under curves, among other applications in mathematics and engineering. The integrand, \(\sin^7 \theta \cos^5 \theta\), indicates that both sine and cosine functions are raised to powers and multiplied together within the integral. Solving this type of integral often involves using trigonometric identities or substitution methods to simplify the expression.
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