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Chapter 5: Problem 5

When using a change of variables \(u=g(x)\) to evaluate the definite integral \(\int_{a}^{b} f(g(x)) g^{\prime}(x) d x,\) how are the limits of integration transformed?

Short Answer

Expert verified
Answer: To transform the limits of integration, we need the original limits (a and b) and the substitution function (g(x)). Then, we plug the original limits into the substitution function to obtain the new limits of integration (u鈧� = g(a) and u鈧� = g(b)).

Step by step solution

01

Identify the substitution

Based on the given information, we will use the substitution \(u = g(x)\). This substitution will allow us to rewrite the original integral in terms of a new variable, \(u\).
02

Differentiate the substitution function

Now, in order to express the integral in terms of \(u\), we should differentiate \(g(x)\) with respect to \(x\): \(\frac{d u}{d x} = g'(x)\). We can then rewrite \(dx\) in terms of \(du\) by solving for \(dx\): \(dx = \frac{du}{g'(x)}\).
03

Transform the limits of integration

Now let's find the new limits of integration by plugging the original limits \(a\) and \(b\) into the substitution function \(g(x)\): \(u_{1} = g(a)\), and \(u_{2} = g(b)\). These are the new limits of integration when the integral is expressed in terms of \(u\).
04

Rewrite the integral in terms of \(u\)

Now, let's rewrite the given integral in terms of \(u\) using the changed limits of integration obtained in Step 3: \(\int_{a}^{b} f(g(x)) g^{\prime}(x) dx = \int_{u_{1}}^{u_{2}} f(u) du\). With the new integral, the problem is simpler, and we can proceed to evaluate it using familiar techniques.

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Most popular questions from this chapter

The population of a culture of bacteria has a growth rate given by \(p^{\prime}(t)=\frac{200}{(t+1)^{r}}\) bacteria per hour, for \(t \geq 0,\) where \(r>1\) is a real number. The increase in the population over the time interval \([0, t]\) is given by \(\int_{0}^{t} p^{\prime}(s) d s\). (Note that the growth rate decreases in time, reflecting competition for space and food.) a. Using the population model with \(r=2,\) what is the increase in the population over the time interval \(0 \leq t \leq 4 ?\) b. Using the population model with \(r=3,\) what is the increase in the population over the time interval \(0 \leq t \leq 6 ?\) c. Let \(\Delta P\) be the increase in the population over a fixed time interval \([0, T] .\) For fixed \(T,\) does \(\Delta P\) increase or decrease with the parameter \(r ?\) Explain. d. A lab technician measures an increase in the population of 350 bacteria over the 10 -hr period [0,10] . Estimate the value of \(r\) that best fits this data point. e. Looking ahead: Work with the population model using \(r=3\) in part (b) and find the increase in population over the time interval \([0, T],\) for any \(T>0 .\) If the culture is allowed to grow indefinitely \((T \rightarrow \infty),\) does the bacteria population increase without bound? Or does it approach a finite limit?

More than one way Occasionally, two different substitutions do the job. Use both of the given substitutions to evaluate the following integrals. $$\int \sec ^{3} \theta \tan \theta d \theta \quad(u=\cos \theta \text { and } u=\sec \theta)$$

Find the area of the following regions. The region bounded by the graph of \(f(x)=x \sin \left(x^{2}\right)\) and the \(x\) -axis between \(x=0\) and \(x=\sqrt{\pi}\).

More than one way Occasionally, two different substitutions do the job. Use both of the given substitutions to evaluate the following integrals. $$\int_{0}^{1} x \sqrt{x+a} d x ; a>0 \quad(u=\sqrt{x+a} \text { and } u=x+a)$$

Consider the function g. which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x} \sin ^{2} t d t$$
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