Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.3.71

Chapter 5, Problem 5.3.71

Areas of regions Find the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval.

ƒ(𝓍) = sin 𝓍 on [―π/4, 3π/4]

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the area of the region bounded by the graph of ƒ(𝓍) = sin(𝓍) and the 𝓍-axis over the interval [―π/4, 3π/4]. This involves integrating the function ƒ(𝓍) = sin(𝓍) over the given interval.

Step 2: Set up the definite integral. The area can be calculated using the formula:

\int a_{- b} \sin dx

Step 3: Evaluate the integral of sin(𝓍). Recall that the integral of sin(𝓍) is −cos(𝓍). Substitute this into the integral:

\int a_{- b} \sin dx

Step 4: Apply the Fundamental Theorem of Calculus. Evaluate −cos(𝓍) at the bounds of the interval [―π/4, 3π/4]. This means substituting the upper bound (3π/4) and the lower bound (―π/4) into −cos(𝓍) and subtracting the results.

Step 5: Simplify the expression. Compute the values of −cos(3π/4) and −cos(―π/4), then subtract them to find the area. Remember that the area is always positive, so take the absolute value if necessary.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

The definite integral of a function over a specified interval represents the net area between the graph of the function and the x-axis. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. In this context, the area can be found by integrating the function ƒ(𝓍) = sin 𝓍 from the lower limit of -π/4 to the upper limit of 3π/4.

Area Under the Curve

The area under the curve of a function can be positive or negative depending on whether the function is above or below the x-axis. When calculating the total area, it is important to consider the absolute value of the areas where the function is negative, as these contribute to the overall area of the region bounded by the graph and the x-axis.

Properties of the Sine Function

The sine function, ƒ(𝓍) = sin 𝓍, is periodic and oscillates between -1 and 1. Understanding its behavior over the interval [-π/4, 3π/4] is crucial for determining the area, as it crosses the x-axis at specific points. This knowledge helps in identifying the segments of the interval where the function is positive or negative, which is essential for accurate area calculation.

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Textbook Question

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍

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Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

∫₋₁² ( ―|𝓍| ) d𝓍

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Textbook Question

Cubic zero net area Consider the graph of the cubic y = 𝓍 (𝓍― a) (𝓍― b), where 0 < a < b. Verify that the graph bounds a region above the 𝓍-axis, for 0 < 𝓍 < a , and bounds a region below the 𝓍-axis, for a < 𝓍 < b. What is the relationship between a and b if the areas of these two regions are equal?

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Textbook Question

Let ƒ(𝓍) = c, where c is a positive constant. Explain why an area function of ƒ is an increasing function.

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Textbook Question

Is x¹² an even or odd function? Is sin x² an even or odd function?

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Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ (x² + x³) dx

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