/*! 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 40 Find the volume of the solid tha... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 40

Find the volume of the solid that is generated when the given region is revolved as described. The region bounded by \(f(x)=\sin x\) and the \(x\) -axis on \([0, \pi]\) is revolved about the \(y\) -axis.

Short Answer

Expert verified
Question: Determine the volume of the solid generated by revolving the region bounded by \(f(x) = \sin x\) and the \(x\)-axis on the interval \([0, \pi]\) about the \(y\)-axis. Answer: The volume of the solid is \(V = 2\pi(\pi-1)\).

Step by step solution

01

Setup the integral for the volume of the solid using the method of cylindrical shells

The volume of the solid can be obtained using the method of cylindrical shells: \(V = 2\pi \int_a^b xy(x) \, dx\) In our problem, we have \(y(x) = f(x) = \sin x\), \(a = 0\), and \(b = \pi\). Hence, we get: \(V = 2\pi \int_0^{\pi} x\sin x \, dx\)
02

Integrate using integration by parts

To evaluate the integral \(\int x\sin x \, dx\), we can use integration by parts, with \(u=x\) and \(dv=\sin x\, dx\). First, let's find the derivatives and integrals of \(u\) and \(dv\): \(du = dx\) \(v = \int \sin x\, dx = -\cos x\) Now apply the integration by parts formula: \(\int x\sin x\, dx = -x\cos x + \int \cos x \, dx\) The last integral is easy to compute: \(\int \cos x\, dx = \sin x\) Thus, we obtain: \(\int x\sin x\, dx = -x\cos x + \sin x + C\)
03

Evaluate the definite integral

Now let's compute the definite integral: \(V = 2\pi \left[ -x\cos x + \sin x \right]_0^\pi = 2\pi \left[ (-\pi\cos\pi + \sin\pi) - (-0\cos 0 + \sin 0) \right]\) \(V = 2\pi \left[ -\pi+1- (0) \right] = 2\pi(\pi-1)\) So the volume of the solid is \(V = 2\pi(\pi-1)\).

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Most popular questions from this chapter

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{d x}{(x+1)\left(x^{2}+2 x+2\right)^{2}}$$

Many methods needed Show that \(\int_{0}^{\infty} \frac{\sqrt{x} \ln x}{(1+x)^{2}} d x=\pi\) in the following steps. a. Integrate by parts with \(u=\sqrt{x} \ln x.\) b. Change variables by letting \(y=1 / x.\) c. Show that \(\int_{0}^{1} \frac{\ln x}{\sqrt{x}(1+x)} d x=-\int_{1}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} d x\) and conclude that \(\int_{0}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} d x=0.\) d. Evaluate the remaining integral using the change of variables \(z=\sqrt{x}\) (Source: Mathematics Magazine 59, No. 1 (February 1986): 49).

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