/*! 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 16 Evaluate the following iterated ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 16

Evaluate the following iterated integrals. $$\int_{0}^{\pi / 4} \int_{0}^{3} r \sec \theta d r d \theta$$

Short Answer

Expert verified
Based on the step by step solution above, evaluate the iterated integral $$\int_{0}^{\frac{\pi}{4}} \int_{0}^{3} r \sec \theta \,dr\,d\theta$$ Solution: The numerical value of the iterated integral is $$\frac{9}{2} \ln(\sqrt{2}+1)$$

Step by step solution

01

Identify the order of integration

The given iterated integral is in the order of integration \(\iint d rd\theta\). The first integral will be evaluated with respect to \(r\), and the second integral will be evaluated with respect to \(\theta\).
02

Evaluate the integral with respect to \(r\)

Let's evaluate the inner integral: $$\int_{0}^{3} r \sec \theta dr$$ To evaluate this integral, we can rewrite it as: $$\sec \theta \int_{0}^{3} r dr$$ Since \(\sec \theta\) is constant with respect to \(r\), we can take it outside the integral. Now, integrate \(r\) with respect to \(r\): $$\sec \theta \left[ \frac{1}{2} r^2 \right]_0^3$$ Now, substitute the limits to get: $$\sec \theta \left[ \frac{1}{2} (3^2 - 0^2) \right] = \frac{9}{2} \sec \theta$$
03

Substitute the result of step 2 into the original integral

Now that we have the result from step 2, which is \(\frac{9}{2} \sec \theta\), we substitute this into the outer integral: $$\int_{0}^{\frac{\pi}{4}} \frac{9}{2} \sec \theta d\theta$$
04

Evaluate the integral with respect to \(\theta\)

Now, we need to evaluate the integral with respect to \(\theta\): $$\int_{0}^{\frac{\pi}{4}} \frac{9}{2} \sec \theta d\theta$$ We can take out the constant term \(\frac{9}{2}\) and rewrite the integral as: $$\frac{9}{2} \int_{0}^{\frac{\pi}{4}} \sec \theta d\theta$$ Now, integrate \(\sec \theta\) with respect to \(\theta\): $$\frac{9}{2} \left[ \ln | \sec \theta + \tan \theta | \right]_0^{\frac{\pi}{4}}$$
05

Substitute the limits into the integral

Now, substitute the limits of integration to find the final answer: $$\frac{9}{2} \left[ \ln | \sec \frac{\pi}{4} + \tan \frac{\pi}{4} | - \ln | \sec 0 + \tan 0 | \right]$$ Now, simplify to get: $$\frac{9}{2} \left[ \ln | \sqrt{2} + 1 | - \ln | 1 | \right]$$ Finally, simplify the expression to get the final answer: $$\boxed{\frac{9}{2} \ln(\sqrt{2}+1)}$$

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