91Ó°ÊÓ

Show that if \(f\) and \(g\) have continuous sceond derivatives and \(f(a)=g(a)=f(b)=g(b)=0,\) then $$\int_{a}^{b} f(x) y^{\prime \prime}(x) d x=\int_{a}^{b} g(x) f^{\prime \prime}(x) d x$$

You are familiar with the identity $$f(b)-f(a)=\int_{a}^{b} f^{\prime}(x) d x$$ (a) Assume that \(f\) has a continuous second derivative. Use integration by parts to derive the identity $$f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(x)(x-b) d x$$ (b) Assume that \(\int\) has a continuous third derivative. Use the result in part (a) and integration by parts to derive the identity $$f(b)-f(a)=f^{\prime}(a)(b-a)+\frac{f^{\prime \prime}(a)}{2}(b-a)^{2}$$ $$-\int_{a}^{b} \frac{f^{\prime \prime \prime}(x)}{2}(x-b)^{2} d x$$ Going on in this manner, we are led to what are called Taylor series (Chapter 12 ).

Use a graphing utility to draw the curve \(y=x \sin x\) for \(x \geq 0 .\) Then use a CAS to calculate the area between the curve and the \(x\) -axis (a) from \(x=0\) to \(x=\pi\) (b) from \(x=\pi\) to \(x=2 \pi\) (c) \(\operatorname{from} x=2 \pi\) to \(x=3 \pi\) (d) What is the area between the curve and the \(x\) -axis from \(x=n \pi\) to \(x=(n+1) \pi ?\) Take \(n\) an arbitrary nonncgative integer.

Use a graphing utility to draw the curve \(y=x \cos x\) for \(x \geq 0 .\) Then use a CAS to calculate the area between the curve and the \(x\) -axis (a) from \(x=\frac{1}{2} \pi\) to \(x=\frac{3}{2} \pi\) (b) \(\operatorname{from} x=\frac{3}{2} \pi\) to \(x=\frac{5}{2} \pi\) (c) from \(x=\frac{5}{2} \pi\) to \(x=\frac{7}{2} \pi\) (d) What is the area between the curve and the \(x\) -axis from \(x=\frac{1}{2}(2 n-1) \pi\) to \(x=\frac{1}{2}(2 n+1) \pi ?\) Take \(n\) an arbitrary positive integer.

Use a graphing utility to draw the curve \(y=1-\sin x\) from \(x=0\) to \(x=\pi .\) Then use a CAS (a) to find the area of the region \(\Omega\) between the curve ard the \(x\) -axis. (b) to find the volume of the solid generated by revolving \(\Omega\) about the \(y\) -axis. (c) to find the centroid of \(\Omega\)