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Chapter 8: Problem 43

Area Find the area of the region bounded by the graphs of \(y=12 /\left(x^{2}+5 x+6\right), y=0, x=0,\) and \(x=1\)

Short Answer

Expert verified
The area of the region bounded by the graphs is \(-6 \ln 2\) square units.

Step by step solution

01

Identify the Integral to Compute

The definite integral of \(y = \frac{12}{{x^2 + 5x + 6}}\) from 0 to 1 over the x-axis should be calculated. This will yield the area of the region bounded by the graphs of the equations.
02

Factorize the Denominator

If possible, factor the denominator \(x^2 + 5x + 6\). The factored form is \((x+2)(x+3)\). The integral expression becomes: \(\int_{0}^{1} \frac{12}{(x+2)(x+3)} dx \)
03

Perform Partial Fraction Decomposition

Next, perform partial fraction decomposition of \(\frac{12}{(x+2)(x+3)}\). This will convert the complex fraction into simpler fractions that can be integrated easily. It is found that \(\frac{12}{(x+2)(x+3)} = \frac{6}{x+2} - \frac{6}{x+3}\)
04

Compute the Integral

With the expression simplified, compute the integral over the interval from 0 to 1: \(\int_{0}^{1} (\frac{6}{x+2} - \frac{6}{x+3}) dx \). This simplifies to \([6 \ln |x+2| - 6 \ln |x+3|]_{0}^{1}\)
05

Evaluate the Integral

Lastly, substitute the limits of the integral to acquire the area: \([6 \ln 3 - 6 \ln 4] - [6 \ln 2 - 6 \ln 3]\). Simplifying this yields \(6 \ln 2 - 6 \ln 4 = -6 \ln 2\)

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Most popular questions from this chapter

Asymptotes and Relative Extrema In Exercises \(75-78\) , find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in previous exercises.) $$ y=\frac{\ln x}{x} $$

Continuous Function In Exercises 101 and \(102,\) find the value of \(c\) that makes the function continuous at \(x=0\) . $$ f(x)=\left\\{\begin{array}{ll}{\left(e^{x}+x\right)^{1 / x},} & {x \neq 0} \\\ {c,} & {x=0}\end{array}\right. $$

Exploration Consider the integral $$\int_{0}^{\pi / 2} \frac{4}{1+(\tan x)^{n}} d x$$ where \(n\) is a positive integer. (a) Is the integral improper? Explain. (b) Use a graphing utility to graph the integrand for \(n=2,4\) \(8,\) and \(12 .\) (c) Use the graphs to approximate the integral as \(n \rightarrow \infty\) . (d) Use a computer algebra system to evaluate the integral for the values of \(n\) in part (b). Make a conjecture about the value of the integral for any positive integer \(n .\) Compare your results with your answer in part (c).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t .\) The Laplace Transform of \(f(t)\) is defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ when the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises \(95-102,\) find the Laplace Transform of the function. $$ f(t)=t $$

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t .\) The Laplace Transform of \(f(t)\) is defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ when the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises \(95-102,\) find the Laplace Transform of the function. $$ f(t)=e^{a t} $$
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