Find the area bounded by $f (x) = sin x$ $f(x)=sinx$ and $g (x) = cos x$ $g(x)=cosx$ from $x = \frac{π}{4}$ $x=pi/4$ to $x = (\frac{5 π}{4} .$ $x=((5pi)/4.$ . Make an accurate sketch of the graphs on the axis below?

Question

Find the area bounded by $f (x) = sin x$ $f(x)=sinx$ and $g (x) = cos x$ $g(x)=cosx$ from $x = \frac{π}{4}$ $x=pi/4$ to $x = (\frac{5 π}{4} .$ $x=((5pi)/4.$ . Make an accurate sketch of the graphs on the axis below?

Find the area bounded by $f (x) = sin x$ $f(x)=sinx$ and $g (x) = cos x$ $g(x)=cosx$ from $x = \frac{π}{4}$ $x=pi/4$ to $x = (\frac{5 π}{4})$ $x=((5pi)/4)$ . Make an accurate sketch of the graphs on the axis below.

1 Answer

Impact of this question

30553 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-30T04:03:14

$A = \int_{\frac{π}{4}}^{5 \frac{π}{4}} sin x - cos x \cdot d x = 2 \sqrt{2}$ $color(blue)[A=int_(pi/4)^(5pi/4)sinx-cosx*dx=2sqrt2]$

Explanation:

The Area due to $x-axis$ $"x-axis"$ between two curves given by:

$A = \int_{a}^{b} y_{2} - y_{1} \cdot d x$ $color(red)[A=int_a^by_2-y_1*dx$

The interval of our integral $x \in [\frac{π}{4}, 5 \frac{π}{4}]$ $x in [pi/4,5pi/4]$

now let set up the integral:

$A = \int_{\frac{π}{4}}^{5 \frac{π}{4}} sin x - cos x \cdot d x = {[- cos x - sin x]}_{\frac{π}{4}}^{5 \frac{π}{4}}$ $A=int_(pi/4)^(5pi/4)sinx-cosx*dx=[-cosx-sinx]_(pi/4)^(5pi/4)$

$[(- sin (5 \frac{π}{4}) - cos (5 \frac{π}{4})) - (- sin (\frac{π}{4}) - cos (\frac{π}{4}))]$ $[(-sin(5pi/4)-cos(5pi/4))-(-sin(pi/4)-cos(pi/4))]$

$[\frac{2}{\sqrt{2}} + \frac{2}{\sqrt{2}}] = \frac{4}{\sqrt{2}} = 2 \sqrt{2}$ $[2/sqrt2+2/sqrt2]=4/sqrt2=2sqrt2$

show the wanted area below(shaded):
$y = sin x$ $y=sinx$ blue curve
$y = cos x$ $y=cosx$ green curve
James

Science

Math

Humanities

... and beyond

Find the area bounded by $f (x) = sin x$ $f(x)=sinx$ and $g (x) = cos x$ $g(x)=cosx$ from $x = \frac{π}{4}$ $x=pi/4$ to $x = (\frac{5 π}{4} .$ $x=((5pi)/4.$ . Make an accurate sketch of the graphs on the axis below?