Question

True or False The area of the region in the first quadrant enclosed by the graphs of \(y=\cos x, y=x,$$ and the $$y$$ -axis is given by the definite integral \)\int_{0}^{0.739}(x-\cos x) d x .$$ Justify your answer.

Short answer

True

Step-by-step solution

Identify the Bounds

The bounds of integration can be found when the two functions intersect, as this forms the enclosed region. Setting \(x = \cos x\) to find these intersection points gives an equation that is best solved by trial and error or numerical methods. A well-known solution to this equation is \(x = 0.739\) (which is known as the Dottie number). Therefore, our bounds of integration are 0 and 0.739.

Evaluate the Integral

To evaluate this integral, remember that the integral of the difference is the difference of the integrals, so split the integral into two: \(\int_{0}^{0.739} x dx - \int_{0}^{0.739} \cos x dx\). The first of these integrals can be evaluated using the power rule and the second using the fundamental theorem of calculus, to get \[ [\frac{1}{2}x^2]^{0.739}_0 - [\sin x]^{0.739}_0 \]. Evaluate this expression to get the numerical value.

Conclusion

The numerical value found is the area under the curve \(y=x\) over the interval [0, 0.739] minus the area under the curve \(y=\cos x\) over the same interval. This is the area of the region described in the problem, and equals to the value of the given definite integral. So the original statement is True.