/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 7

Evaluate the following integrals or state that they diverge. $$\int_{-\infty}^{0} e^{x} d x$$

Short Answer

Expert verified
If it converges, what is its value? Answer: The integral converges, and its value is 1.

Step by step solution

01

Identify the integrand and the limits of integration

The function we need to integrate is \(e^x\), and the given limits of integration are from \(-\infty\) (negative infinity) to 0.
02

Find the antiderivative of the integrand

The antiderivative of \(e^x\) is itself, that is, the indefinite integral of \(e^x\) with respect to x is \(e^x + C\), where C is the constant of integration. We don't need to worry about the constant of integration for now, as it will be eliminated when we evaluate the definite integral.
03

Evaluate the definite integral

Now, we need to evaluate the integral of \(e^x\) with the given limits of integration. This can be done by finding the difference between the antiderivative at the upper and lower limits of integration as follows: $$\int_{-\infty}^{0} e^x dx = \left[e^x\right]_{-\infty}^{0}$$
04

Calculate the integral at the limits of integration

To find the integral at the limits of integration, we simply plug in the limits into the antiderivative: At the upper limit, \(x=0\): $$e^0 = 1$$ At the lower limit, \(x=-\infty\): We know that as \(x\) goes to \(-\infty\), \(e^x\) goes to 0. Therefore, the lower limit evaluates to 0: $$\lim_{x\to-\infty} e^x = 0$$
05

Subtract the value of the antiderivative at the lower limit from the value at the upper limit

Finally, we find the difference between the value of the antiderivative at the upper limit and the lower limit: $$\int_{-\infty}^{0} e^x dx = 1 - 0 = 1$$ The integral converges to 1.

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Most popular questions from this chapter

Apply Simpson's Rule to the following integrals. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(7 .\) Make \(a\) table similar to Table 7.8 showing the approximations and errors for \(n=4,8,16,\) and \(32 .\) The exact values of the integrals are given for computing the error. \(\int_{0}^{4}\left(3 x^{5}-8 x^{3}\right) d x=1536\)

Evaluate the following integrals. Assume a and b are real numbers and \(n\) is an integer. $$\int \frac{x}{a x+b} d x \text { (Use } u=a x+b$$

The region bounded by \(f(x)=(4-x)^{-1 / 3}\) and the \(x\) -axis on the interval [0,4) is revolved about the \(y\) -axis. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(x\) -axis on the interval (-1,1] is revolved about the line \(y=-1\).

An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem \(B^{\prime}(t)=a B-m\) for \(t \geq 0,\) with \(B(0)=B_{0} .\) The constant \(a\) reflects the annual interest rate, \(m\) is the annual rate of withdrawal, and \(B_{0}\) is the initial balance in the account. a. Solve the initial value problem with a=0.05, m= 1000 dollar \(/\mathrm{yr}\), and \(B_{0}\)= 15,000 dollar. Does the balance in the account increase or decrease? b. If \(a=0.05\) and \(B_{0}\)= 50,000 dollar, what is the annual withdrawal rate \(m\) that ensures a constant balance in the account? What is the constant balance?

Show that \(L=\lim _{n \rightarrow \infty}\left(\frac{1}{n} \ln n !-\ln n\right)=-1\) in the following steps. a. Note that \(n !=n(n-1)(n-2) \cdots 1\) and use \(\ln (a b)=\ln a+\ln b\) to show that $$ \begin{aligned} L &=\lim _{n \rightarrow \infty}\left(\left(\frac{1}{n} \sum_{k=1}^{n} \ln k\right)-\ln n\right) \\ &=\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \frac{k}{n} \end{aligned} $$ b. Identify the limit of this sum as a Riemann sum for \(\int_{0}^{1} \ln x d x\) Integrate this improper integral by parts and reach the desired conclusion.
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