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(a) Graph the functions $ f(x) = \frac{1}{x^{1.1}} $ and $ g(x) = \frac{1}{x^{0.9}} $ in the viewing rectangles $ [0, 10] $ by $ [0, 1] $ and $ [0, 100] $ by $ [0, 1] $.(b) Find the areas under the graphs of $ f $ and $ g $ from $ x = 1 $ to $ x = t $ and evaluate for $ t = 10, 100, 10^4 , 10^6 , 10^{10} $, and $ 10^{20} $.(c) Find the total area under each curve for $ x \ge 1 $, if it exists.
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 8
Improper Integrals
Integration Techniques
Campbell University
Oregon State University
Baylor University
Lectures
01:53
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
27:53
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
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(a) Graph the functions $f…
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Sketch the graphs of the f…
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Draw the graphs of $f(x)=…
For this problem, i just give you some ideas about how to do the part b and part c for part b, integral of f x from 1 to t x, but computation. This is equal to integral from 1 to t x, to make 2.1 1.1 x, and this is equal to negative 10 times x, 2 negative 0 on 1 from 1 to t point and we can write it. This is 10 times 1 minus t to negative 0.1 and then just plug in 1000 and out i numbers and similarly integral of g x. This is equal to is equal to 10 times x, 20.1 from 1 to t. This is equal to 10 times to 0.1 minus 1 point now sore plotting those numbers. This is part b for part c. The total area in the h curve for x greater than 1 is equal to for the function f x. This is equal to the limit. He goes to infinity of the integral of 1 to t of the function f x, d x and when t goes to infinity, he to negative 0.1 goes to 0, so the limit is equal to 10 and for the function g will wax limit. He goes to infinity integral 1 to t was the function godley can see when t goes to infinity to the o. Power goes to infinity, so this is averted.
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