Question

Determine whether the following statements are true and give an explanation or counterexample.b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

Accepted Answer

First, identify the two functions given: $$y = \sin x$$ and $$y = \cos x on $$the interval $$[0, \frac{\pi}{2}]. $$Determine which function is on top (greater) and which is on the bottom over the interval $$[0, \frac{\pi}{2}]. $$This is important because the area between two curves is found by integrating the difference between the top function and the bottom function. Evaluate the values of $$\sin x$$ and $$\cos x at $$the endpoints: at $$x=0$$, $$\sin 0 = 0$$ and $$\cos 0 = 1$$, so $$\cos x is $$greater; at $$x=\frac{\pi}{2}$$, $$\sin \frac{\pi}{2} = 1$$ and $$\cos \frac{\pi}{2} = 0$$, so $$\sin x is $$greater. This means the two curves cross somewhere in the interval. Find the point where $$\sin x = \cos x in$$ $$[0, \frac{\pi}{2}] by $$solving $$\sin x = \cos x. $$This will give the exact point where the top and bottom functions switch. To find the total area between the curves, split the integral at the crossing point and integrate the absolute difference accordingly: $$\$$int_0$$^{c} (\cos x - \sin x) \, dx + \int_c$$^{\frac{\pi}$${2}} (\sin x - \cos x) \, dx$$, where $$c is $$the crossing point. Therefore, the statement that the area is $$\$$int_0$$^{\frac{\pi}{2}} (\cos x - \sin x) \, dx$$ without splitting is not correct.

Determine whether the following statements are true and give an explanation or counterexample.

b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

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Key Concepts

Definite Integral as Area Between Curves

Determining Which Function is on Top

Evaluating the Integral and Sign of the Result