Textbook Question
Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(a) Evaluate H (0) .
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8. Definite Integrals
Introduction to Definite Integrals
4:11 minutesProblem 8.R.125a
Textbook Question
125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.
a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.
Verified step by step guidance1
Step 1: Begin by recalling the reduction formula for integrals involving powers of trigonometric functions. The goal is to express the integral of sin^m(x) in terms of the integral of sin^(m−2)(x). Start with the integral I = ∫ from 0 to π of (sin^m(x)) dx.
Step 2: Use integration by parts to simplify the integral. Let u = sin^(m−1)(x) and dv = sin(x) dx. Then, compute du = (m−1)sin^(m−2)(x)cos(x) dx and v = −cos(x). Substitute these into the integration by parts formula: ∫ u dv = uv − ∫ v du.
Step 3: Evaluate the boundary terms uv at x = 0 and x = π. Note that sin(0) = sin(π) = 0, so the boundary terms vanish. This simplifies the integral to −∫ from 0 to π of (−cos(x) * (m−1)sin^(m−2)(x)cos(x)) dx.
Step 4: Simplify the remaining integral. Combine the cos^2(x) term and use the Pythagorean identity cos^2(x) = 1 − sin^2(x) to express the integral in terms of sin^(m−2)(x). This leads to a recursive relationship between the integrals of sin^m(x) and sin^(m−2)(x).
Step 5: Factor out constants and simplify the recursive formula to show that ∫ from 0 to π of (sin^m(x)) dx = (m−1)/m × ∫ from 0 to π of (sin^(m−2)(x)) dx. This completes the proof of the reduction formula.
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4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reduction Formula
A reduction formula is a recursive relationship that expresses an integral in terms of another integral with a lower power or degree. In this context, it allows us to simplify the computation of integrals involving powers of sine by relating them to integrals of lower powers, making it easier to evaluate complex integrals step by step.
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Recursive Formulas
Definite Integral
A definite integral calculates the accumulation of a quantity, represented as the area under a curve, between two specified limits. In this case, the integral from 0 to π of sin^m x provides a specific numerical value that represents the area under the curve of sin^m x over the interval [0, π], which is crucial for applying the reduction formula.
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Wallis Product
The Wallis product is an infinite product that relates to the values of sine and cosine functions and is used to derive various results in calculus, particularly in evaluating integrals. It is significant in the context of this problem as it connects the reduction formula to the broader implications of sine integrals and their convergence properties.
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