Blog: `y = cos(x), y = x + 2sin^4 (x)` Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

Saturday, July 24, 2010

$\displaystyle{y}={\cos{{\left({x}\right)}}},{y}={x}+{2}{{\sin}}^{{4}}{\left({x}\right)}$ Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

To graph the area between curves, we determine the boundary values based on the intersection points.

In the xy-plane, the graph of $\displaystyle{y}={\cos{{\left({x}\right)}}}$ is plotted in red color while the graph of $\displaystyle{y}={x}+{2}{{\sin}}^{{{4}}}{\left({x}\right)}$ is plotted in blue color.

Using TI-84 calculator, you can solve for the intersection points as:

(-1.911917,-0.3345439), (-1.223676,0.34019093), and (0.60794574,0.82082311).

To solve for these points, use the option Calc/5: intersect in your TI-84 calculator.

You may position the cursor along the blue graph for the first curve and let the cursor be along the red graph for the second curve. The cursor should be near or before the intersection point you want to solve. Then press Enter for the Guess. X=# and Y=# will appear below your calculator screen for the ordered pair of a certain intersection point. Repeat the steps until you have determined the three intersection points.

The x-values from the intersection points will be used as the boundary values of your integral or the limits of integration.

The formula to solve the bounded area between two curves follows:

A = $\displaystyle{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$

where $\displaystyle{f{{\left({x}\right)}}}={y}_{{{a}{b}{o}{v}{e}}}-{y}_{{{b}{e}{l}{o}{w}}}$

or $\displaystyle{f{{\left({x}\right)}}}={y}_{{{u}{p}{p}{e}{r}}}-{y}_{{{l}{o}{w}{e}{r}}}$

It can also be written as A = $\displaystyle{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}{\left.{d}{x}\right.}$

such that $\displaystyle{f{{\left({x}\right)}}}\geq{g{{\left({x}\right)}}}$ on the interval [a,b].

For the first bounded region, the upper curve is $\displaystyle{y}={x}+{2}{{\sin}}^{{{4}}}{\left({x}\right)}$ and the lower curve is $\displaystyle{y}={\cos{{\left({x}\right)}}}$ from x= -1.911917 to x=-1.223676.

$\displaystyle{A}_{{1}}=$ $\displaystyle{\int_{{-{{1.911917}}}}^{{-{{1.223676}}}}}{\left[{x}+{2}{{\sin}}^{{{4}}}{\left({x}\right)}-{\cos{{\left({x}\right)}}}\right]}{\left.{d}{x}\right.}$

$\displaystyle{A}_{{1}}={0.1949657715}\approx{0.19497}$

For the second bounded region, the upper curve is y=cos(x) and the lower curve is $\displaystyle{y}={x}+{2}{{\sin}}^{{{4}}}{\left({x}\right)}$ from x=-1.223676 to x=0.60794574.

$\displaystyle{A}_{{2}}=$ $\displaystyle{\int_{{-{{1.223676}}}}^{{0.60794574}}}{\left[{\cos{{\left({x}\right)}}}-{x}-{2}{{\sin}}^{{{4}}}{\left({x}\right)}\right]}{\left.{d}{x}\right.}$