$\frac{h}{\left(\frac{\sin \left(x\right)}{2}\right)}\=\frac{\sqrt{3}}{1}\Rightarrow$$h\=\frac{\sqrt{3}}{2}\sin \left(x\right)$

$V\={\int }_0^{\mathrm{\π}\/2}A\left(x\right) \mathit{\ⅆ}x$ = $\frac{\sqrt{3}}{4}{\int }_0^{\mathrm{\π}\/2}{\sin }^2\left(x\right) \mathit{\ⅆ}x\=\frac{\mathrm{\π} \sqrt{3}}{16}$ 

$\frac{\sqrt{3}}{4}{\int }_0^{\mathrm{\π}\/2}{\sin }^2\left(x\right) \mathit{\ⅆ}x$ = $\frac{1}{16}\sqrt{3}\mathrm{\π}$$$$$

Stepwise evaluation of the integral

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${\int }_0^{\mathrm{\pi }\/2}{\sin }^2\left(x\right) \mathit{\ⅆ}x$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Integration Steps}\\ & & {\int }_0^{\frac{\mathrm{\pi }}{2}}{\sin \left(x\right)}^2\ⅆx\\ \text{▫}& & \text{1. Rewrite}\\ & \text{◦}& \text{Equivalent expression}\\ & & {\sin \left(x\right)}^2=\frac{1}{2}-\frac{\cos \left(2x\right)}{2}\\ & & \text{This gives:}\\ & & {\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\frac{1}{2}-\frac{\cos \left(2x\right)}{2}\right)\ⅆx\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{sum}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ & & \int \left(f\left(x\right)+g\left(x\right)\right)\ⅆx=\int f\left(x\right)\ⅆx+\int g\left(x\right)\ⅆx\\ & & f\left(x\right)=\frac{1}{2}\\ & & g\left(x\right)=-\frac{\cos \left(2x\right)}{2}\\ & & \text{This gives:}\\ & & {\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{1}{2}\ⅆx+{\int }_0^{\frac{\mathrm{\pi }}{2}}-\frac{\cos \left(2x\right)}{2}\ⅆx\\ \text{▫}& & \text{3. Apply the}\mathbf{\text{constant}}\text{rule to the term}\int \frac{1}{2}\ⅆx\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant}}\text{rule}\\ & & \int C\ⅆx=Cx\\ & \text{◦}& \text{This means}\\ & & \int \frac{1}{2}\ⅆx=\frac{x}{2}\\ & & \text{We can now rewrite the integral as:}\\ & & \frac{\mathrm{\pi }}{4}+{\int }_0^{\frac{\mathrm{\pi }}{2}}-\frac{\cos \left(2x\right)}{2}\ⅆx\\ \text{▫}& & \text{4. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\int -\frac{\cos \left(2x\right)}{2}\ⅆx\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \int Cf\left(x\right)\ⅆx=C\left(\int f\left(x\right)\ⅆx\right)\\ & \text{◦}& \text{This means:}\\ & & \int -\frac{\cos \left(2x\right)}{2}\ⅆx=-\frac{\left(\int \cos \left(2x\right)\ⅆx\right)}{2}\\ & & \text{We can rewrite the integral as:}\\ & & \frac{\mathrm{\pi }}{4}-\frac{\left({\int }_0^{\frac{\mathrm{\pi }}{2}}\cos \left(2x\right)\ⅆx\right)}{2}\\ \text{▫}& & \text{5. Apply a change of variables to rewrite the integral in terms of}u\\ & \text{◦}& \text{Let}u\text{be}\\ & & u=2x\\ & \text{◦}& \text{Isolate equation for}x\\ & & x=\frac{u}{2}\\ & \text{◦}& \text{Differentiate both sides}\\ & & \mathrm{dx}=\frac{\mathrm{du}}{2}\\ & \text{◦}& \text{Substitute the values for x and dx back into the original}\\ & & {\int }_0^{\frac{\mathrm{\pi }}{2}}\cos \left(2x\right)\ⅆx={\int }_0^{\mathrm{\pi }}\frac{\cos \left(u\right)}{2}\ⅆu\\ & & \text{This gives:}\\ & & \frac{\mathrm{\pi }}{4}-\frac{\left({\int }_0^{\mathrm{\pi }}\frac{\cos \left(u\right)}{2}\ⅆu\right)}{2}\\ \text{▫}& & \text{6. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\int \frac{\cos \left(u\right)}{2}\ⅆu\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \int Cf\left(u\right)\ⅆu=C\left(\int f\left(u\right)\ⅆu\right)\\ & \text{◦}& \text{This means:}\\ & & \int \frac{\cos \left(u\right)}{2}\ⅆu=\frac{\left(\int \cos \left(u\right)\ⅆu\right)}{2}\\ & & \text{We can rewrite the integral as:}\\ & & \frac{\mathrm{\pi }}{4}-\frac{\left({\int }_0^{\mathrm{\pi }}\cos \left(u\right)\ⅆu\right)}{4}\\ \text{▫}& & \text{7. Evaluate the integral of}\cos \text{(}u\text{)}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{cos}\text{rule}\\ & & \int \cos \left(u\right)\ⅆu=\sin \left(u\right)\\ & \text{◦}& \text{Apply limits of definite integral}\\ & & \genfrac{}{}{0ex}{}{\sin \left(u\right)}{\phantom{u=\mathrm{\pi }}}|\genfrac{}{}{0ex}{}{\phantom{\sin \left(u\right)}}{u=\mathrm{\pi }}-\left(\genfrac{}{}{0ex}{}{\sin \left(u\right)}{\phantom{u=0}}|\genfrac{}{}{0ex}{}{\phantom{\sin \left(u\right)}}{u=0}\right)\\ & & \text{This gives:}\\ & & \frac{\mathrm{\pi }}{4}\end{array}$

Figure 5.3.3(d)   Context Panel access to Integration Rules