Definite integral word problem with trig. Check setup?

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$y = \sin x$, $y = \cos x$, $0 < x < \frac{\pi}{4}$; rotated about the line $y = -1$

So...

outer radius: $1 + \cos x$

inner radius: $1 + \sin x$

So here's my setup:

$$\pi \int_0^\frac{\pi}{4} (1 + \cos x)^2 - (1 + \sin x)^2 \, dx$$

$$\pi \int_0^\frac{\pi}{4} 1 + 2 \cos x + \cos^2x - ( 1 + 2 \sin x + \sin^2 x ) \, dx$$

Is this setup right?

But then I'm stuck here on the next step:

$$\pi \int_0^{\frac{\pi}{4}} (2 \cos x - 2 \sin x + \cos^2 x + \sin^2 x \,) dx$$

From here, I need to find the antiderivatives. but what are the antiderivatives of $\cos^2 x$? Also, how do I draw this graph using mathjax?

Answer 1

Hint

As pointed out in the comments you have: $$\cos^2(x)-\sin^2(x)=\cos(2x)$$ In all generality when you have $\cos^2$ or $\sin^2$ you can use the formulas: $$\cos^2(x)=\frac{1+\cos(2x)}{2}$$ $$\sin^2(x)=\frac{1-\cos(2x)}{2}$$ from where it is easy to compute anti derivatives.

---

For the graph I don't think there is a easy way to fraw graph using mathjax but in LaTeX you can use pgfplots:

\documentclass{standalone}
\usepackage{pgfplots}


\usepgfplotslibrary{fillbetween}
\usetikzlibrary{patterns}

\begin{document}

\begin{tikzpicture}
\begin{axis}[axis lines=middle,
            xlabel=$x$,
            ylabel=$y$,
            enlargelimits,
            ytick=\empty,
            xtick={0.785},
            xticklabels={$\pi/4$}]
\addplot[name path=S,blue,domain={0:pi/4}] {sin(deg(x))} node[pos=0.5, below]{$y=\sin(x)$};

\addplot[name path=C,red,domain={0:pi/4}] {cos(deg(x))}node[pos=0.5, above]{$y=\cos(x)$};

\addplot[pattern=north west lines, pattern color=black!20]fill between[of=S and C]
;


\end{axis}
\end{tikzpicture}
\end{document}