Textbook Question
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«βΒΉ (πΒ³ β 2π) dπ = β3/4
(a) β«βΒΉ (4πβ2πΒ³) dπ
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.1.69a
Approximating areas Estimate the area of the region bounded by the graph of Ζ(π) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.
(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.
Verified step by step guidance1
First, divide the interval [0, 2] into 4 equal subintervals. Since the length of the interval is 2, each subinterval will have width \(\Delta x = \frac{2 - 0}{4} = 0.5\).
Identify the left endpoints of each subinterval. These will be \(x_0 = 0\), \(x_1 = 0.5\), \(x_2 = 1.0\), and \(x_3 = 1.5\). Note that the left Riemann sum uses the function values at these points.
Evaluate the function \(f(x) = x^2 + 2\) at each left endpoint: calculate \(f(x_0)\), \(f(x_1)\), \(f(x_2)\), and \(f(x_3)\).
Multiply each function value by the width \(\Delta x\) to find the area of each rectangle: \(f(x_i) \times \Delta x\) for \(i = 0, 1, 2, 3\).
Sum all these rectangle areas to approximate the total area under the curve on [0, 2]: \(\text{Left Riemann Sum} = \sum_{i=0}^{3} f(x_i) \Delta x\). This sum represents the approximate area bounded by the graph and the x-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Riemann Sums
Riemann sums approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles formed using function values at specific points. The left Riemann sum uses the left endpoint of each subinterval to determine the rectangle height, providing an estimate of the integral.
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06:11
Introduction to Riemann Sums
Partitioning the Interval
Partitioning involves dividing the interval [0, 2] into equal subintervals, here n = 4, to create smaller segments for approximation. Each subinterval has length Ξx = (b - a)/n, which is essential for calculating the width of rectangles in the Riemann sum.
Recommended video:
08:44Interval of Convergence
Function Evaluation at Endpoints
To compute the left Riemann sum, evaluate the function f(x) = xΒ² + 2 at the left endpoints of each subinterval. These values determine the heights of the rectangles, which when multiplied by the subinterval width and summed, approximate the total area under the curve.
Recommended video:
4:26Evaluating Composed Functions
Related Practice
Textbook Question
Area functions for linear functions Consider the following functions Ζ and real numbers a (see figure).
(a) Find and graph the area function A (π) = β«βΛ£ Ζ(t) dt .
Ζ(t) = 2t + 5 , a = 0
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Textbook Question
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(a) Write the left and right Riemann sums in sigma notation for an arbitrary value of n.
β«βΒΉ cos β»ΒΉ π dπ
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Textbook Question
Average value with a parameter Consider the function Ζ(π) = aπ (1βπ) on the interval [0, 1], where a is a positive real number.
(a) Find the average value of Ζ as a function of a .
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Textbook Question
Working with area functions Consider the function Ζ and the points a, b, and c.
(a) Find the area function A (π) = β«βΛ£ Ζ(t) dt using the Fundamental Theorem.
Ζ(π) = cos π ; a = 0 , b = Ο/2 , c = Ο
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Textbook Question
Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(a) Find the mass of the left half of the rod (0 β€ x β€ 5) .
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